Maritime travel and warfare in the Mediterranean dates back centuries to Roman times. All that hopping around, both trade and warfare, between various places would have built up experience and familiarity with travel by sea, and led to the demand and building of better, bigger, and more capable vessels for progressively longer voyages in more challenging waters. If you needed to get from the "boot" of Italy to, say, Greece or Turkey, far easier to go by sea than by land. China would have little or no such need; land travel would suffice or be the best option between most locations. That disparity in need/benefit and acquired experience (and technological development) enabled the Europeans to undertake the truly long distance voyages around the Horn of Africa to India and China when it became beneficial to bypass the Middle East. It was a progression of smaller steps with worthwhile benefits at each step that ultimately led to the readiness to undertake the big adventures.

Expanding on some of the previous answers:

Europe needed to bypass the Moors, and China was self-sufficient in trade or busy with the Mongols. True, but "self-sufficiency" is also cultural. There may be money or prestige to be gained in foreign trade even if it is not strictly needed. And more gold would help to deal with foreign threats. China would call itself the middle-kingdom, and the outsiders barbarians, tributaries, etc, so foreign deals were not valued. The Europeans valued their missionaries and explorers' adventures, as the abundant contemporary literature on missions and explorations attests.

Christianity also would give other advantages: In the XV century, China certainly were much more advanced in many aspects of navigation. It must have had also a corresponding community involved in trade, shipbuilding, navigation knowledge and logistical support, otherwise Zheng He could not go anywhere. But everything disappeared just because the Emperor decided so! He even ordered whole groups of workers killed!

While it is possible for an European king to be a bloodthirsty tyrant, it would be much harder to adopt radical measures such as killing the workers, just to close an industry that is not even disloyal! It would be seen as unchristian.

It would be also strange for a Christian king, who has the means to reach distant pagan places, and strong hope of profit there, just decide that the country must be closed to foreign trade. It is not just about the money, what about the souls of these pagans? A Chinese emperor could not care less about the souls of distant barbarians.

About Jared Diamond book, its most strong arguments are about the difference between Eurasia x rest of world (America, Sub-Saharian Africa, Australia).

Most of its arguments (N-S x W-E orientation, yield of crops, quality of pack-animals, geographical choke-points, etc), are not as strongly applicable to discriminate between countries in the Eurasia landmass (Europe->Middle East->India->China).

So in Eurasia we must look more strongly to cultural or religious arguments.

Also, people often forget that the Portuguese discoveries were a large multi-generational effort driven by the crown and the Order of Christ since the first half (not the second half) of the XV century. Some interesting factors are in play here:

The Portuguese did not just wanted to get rich bypassing the Moors. They were also afraid of another invasion of the Moors (after all, the Reconquista had to deal with Almohads and Almoravids) They knew that the Moors in Morroco had a profitable trade with saharian and sub-saharian Africa via caravans, and if the Portuguese could bypass this trade, it would financially hurt the Morrocans (as it did). If you look the belligerent text of the papal bull about portuguese navigations, "go, invade and slave the ones helping the enemies of the church" it has this context: stop the revenue flow of your enemies or risk being obliterated. So the trade was also a way to win over an existential threat - China did not had this incentive.

Also, the Order of Christ is very interesting. Yes, it was not anymore a true religious order, its members were not celibate or monks anymore, and the king was the grand-master. But, it still had a nice revenue which they must use to defend the kingdom and spread Christendom. If the king would just pocket the money, it would be an impious loss of face before the nobles of the Order. Directing the money towards the navigations was a means of accomplishing all these goals. The other option was to continue the crusades directly, but as 1580 shows, this was not a good idea and many of them could see that. (BTW, the red cross seen in caravels is a symbol of the Order of Christ, not the Portuguese crown)

So, another religious argument, not just a general missionary spirit, but a direct source of revenue coming from an organization heir to the crusades.

About the other great civilizations:

I think the Hindu religion has something against traveling overseas, which is not so important today but might have been more important before. At least some Hindu friends told me that. And obviously the Muslims also kept them busy, even more than the Mongols kept the Chinese busy...

The Arabs also did trade and sent missionaries. But jihad could be accomplished with armies - they bordered christian and hindu lands. Why would a large state make a difficult effort to navigate to distant lands if the infidel was just at the border?

Setting aside the question of how European ocean travels compares to the rest of the world (see other answers for that), it is also important to distinguish between various parts of Europe.

Many European countries did not develop any major seafaring until the 17th or 18th century. In the case of Russia, for instance, this is simply due to lack of ocean access and it not being a major power player until the 17th century. Even today, Russia is struggling to develop naval power; they simply don't have enough ice-free ports. Murmansk is not ice-free, Vladivostok is at the far end of Asia, and Sevastopol is Ukrainian (although currently annexed by Russia).

The regions that had did dominate the oceans in the 1500s through 1700s or so were Spain/Portugal/Holland, England, and the western part of the Hanseatic region (Germany through Norway).

The very long-distance travel, on a massive (for the standards of the time) scale, is also inextricably linked to colonialism - not just as a cause of colonialism, but also as an effect.

The Hanesatic League evolved from relatively short trade routes in the Baltic Sea, and from a Viking tradition of long-distance seafaring. This region was thriving on mostly internal trade, so they didn't get much into either colonialism, or into the very long-distance ocean travel, even though they were a major seafaring region. By the 1500s, which you asked about, they were also already in decline.

England had to develop maritime strength simply because of its island nature (and not just one island, but a whole archipelago with thousands, from Isle of Man to the Shetlands), and to avoid depending on France as their only close-by access to the continent. Colonialism was then a consequence of the seafaring tradition - might as well use the new maritime strength to try and get rich, while getting rid of criminals at the same time.

The region of Spain etc. (which at the time included much of today's Netherlands) wanted to build a new economic base, by competing with the Arab world on trade with Asia.

So both England and Spain had major incentives for developing maritime culture.

China really didn't have comparable reasons. Their conquests were almost all neighboring countries or very nearby islands.

Of course this is highly simplistic. I'm ignoring the Mediterranean traffic going back to ancient Greece and Egypt entirely, for instance.

In proportion to its area, Europe has a hugely long coastline, with many islands and many natural harbors. The average distance to the nearest seacoast is much smaller in Europe than in China.

European merchants were state backed

A recurring theme in Chinese history is the tension between merchants and mandarins. Merchants make money and bring wealth to China. The mandarins were members of the civil service who studied Confucian doctrine and were naturally wary of merchants.

Of course, there were plenty if cleric-ish types in Europe, as well. The equivalent of the mandarins there was the Catholic church. But there is a significant difference here. The Confucian cultural organization in China was intimately attached to the palace, and that palace controlled the entirety of China, at least in the Ming and early Qing periods from ~1400 to ~1800. In Europe, the Catholic church was the dominant cultural organization at the beginning of this time period, but it was not intimately attached to the political powers of Europe. Surely it was influential, but Kings in Castille, Portugal, France, and England, along with the Holy Roman Emperor, were ultimately independent.

Two events made this clear at the very beginning of the Age of Exploration. The first is the schism of the 14th century. There were for many decades two competeing popes: one in Avignon associated with the French crown, and another in Rome. Kings were free to align with either pope depending on shifting political conditions. This situation lasted until 1417. Almost exactly 100 years later, a more permanent schism happened: the Reformation. This cut papal influence completely out of national politics in England and the Netherlands, in particular.

The result of this disintegration of central authority is that seagoing merchants were able to obtain state support to fund their ventures at the dawn of the Age of Exploration. Recall how Columbus sought the patronage of first the Portuguese, and then the Catholic Monarchs of Spain. Royal patronage was crucial in putting together many of the first expeditions; like Diaz and Da Gama's efforts to reach India, and Columbus and Cabot's trips to the New World.

When one country was less interested in supporting exploration (France never was very interested in exploration, England had a series of internal conflicts to distract it), others were more interested (Portugal in particular, and Spain as well). Later, when the Spanish lost the interest in funding additional exploratory trips, Englishmen like Drake went. When Portuguese investment in the Indian Ocean abated, the Dutch took over their interests.

Meanwhile, in China, which was a unitary country, when the government stopped funding exploration, there were no other governments to turn to.

Cannon were mostly invented in Europe

The other important facet of Europe's naval power is that its ships truly were superior. Much is made of how big Chinese junks were, and how Zheng He's fleet was more impressive than anything the Europeans could do.

Yet, the Battle of Diu fortold the outcome of every naval battle between Europeans and others until Tsushima. In 1509, the Portugese had just broken into the Indian Ocean less than 20 years before. The Muslim powers that were decided to destroy the interlopers and win back the Indian Ocean trade. But the Portuguese crushed a fleet that had perhaps 5:1 numerical superiority; to the point that there were only 32 Portuguese casualties. The key to victory was naval gunnery, something that only Europeans had developed to that point.

One of the great mysteries of history is perhaps why no other part of the world was able to develop naval gunnery to a level remotely matching Europeans. Several areas had the raw materials and maritime tradition to imitate European seamanship: Japan, Fujian and Guangzhou, and South India, in particular. Yet none did.

This is the proximate cause of Europeans dominating the ocean; the more remote causes are probably related to the first section of this answer.

My professional historian girlfriend points out that the question ignores the worldwide Arab traders who dominated ocean traffic, or the Phonecians. I'd add that my understanding is that much of Oceania was settled by Chinese. I think OP's thesis stated more strongly than evidence supports.

I'm grateful to @curiousdanni for the correction:

"I'd add that my understanding is that much of Oceania was settled by Chinese" - correction, by Polynesians, a sub group of Austronesia, whose languages appear to have originated in Taiwan, but from the indigenous Taiwanese (Formosan), not the ethnic Chinese. The Chinese did colonise much of SE Asia though.

OP does not limit the question in time, merely asserts that the dominance is more evident after the 15th century. I responded to the question that OP asked; part of my point was to challenge the frame of many of the other answers.

Complex and disputed social, economical and cultural reasons

With decent technology and a brave and enterpreneurish heart anyone can be a sea explorer.

Leif Erikson from Iceland reached America, Zang He explored the Indian Ocean with chinese ships superior to their european counterparts. Pytheas circumnavigated Britannia in the classical antiquity... I count the Portugese themselves in this category. They hade trade stations form Brazilia to India and tried to divide the globe with the Spanish, but yet in mere decades they lost their leading role.

And jet, to exchange momentary advantage in geological knowledge and navigational skill into a lasting military, economic and political presence - and eventually dominance - more is needed: Some kind of 'Urge' or even 'Greed' in the society itself which drives massive amount of men into the new lands.

At the start of the Viking age the Norse had this. But they run out by the time they reached Vinland and retreated after the first skrimishes with the native people. The countless ships carring the new settlers did not come. They were busy establising Cnut in England or infighting over local kingdoms in Norway. Same for the Greeks. This 'Urge' in the society pushed them to colonize half the Mediterraneum, but was not enough for the Atlantic. I know little about Chinese history, but it was probably same. Their explorers rushed out into the unknown, established long lasting and fruitfull trade relations, but the great expansion did not come. Somehow they remained content with trading silk and spices (and iron, tea, etc...) with the Arabs, their eyes turned inwards.

Why had the early modern Europe more from this 'Urge' than anyone before? This is a very hard question, I can barely do more than outline some vague ideas.

Here is a book from István Hajnal I mentioned. It is quite old (1930s) but it searches the answer preciselly for this question.

Here are some partial answers.

In the early modern period there happened to be an economic system (early capitalism) in some European countries (like England or the Netherlands), that allowed individuals to go exploring 'on will', but also to build on a rich economic and social base. (There were sailors and settlers wishing to go, businessmen investing in things like the East India Company, etc...)

But coincidentally there were powerfull kings too, who could support voyages and develop naval power to protect the merchants and colonial interest.

The HEIC I think is quite a good example. It was always supported by the Kings (and Queens) but it was also a commercial enterprise running for profit and having thousands of employees. At the height of it's power its army was stronger than the british army itself.

Also, somehow Europe managed to use the riches obtained to strenghten its economy and recreate its 'urge': It is almost a cliche how the Spanish wasted the gold and silver obtained from America, making aristocratic courts rich with luxories and doing pious donations. So that their money flowed into other places (where the wares buyed by their nobles were produced)

Here lies the chief contrast with the Vikins. The Norse were quite individualistic (while also family centric) and had a certain 'businesslike' attitude. But their social and economical institutions weren't developed enough, so they could only use their spoils to hoard and show off, and to enjoy its pleasures. They had little means to accumulate capital, so as the primary pressure (overpopulation and poor lands in the heartlands) was released, they come ta a halt.

I think we first have to ask what we mean when we say that the Europeans “dominate[d] oceans.” You describe this as something that “becomes more obvious after the fifteenth century,” when Europe began colonizing and conquering.

However, Europeans did not in any sense dominate the Pacific at this time; the Polynesians colonized New Zealand about two centuries before the start of that period, and had already colonized as far east as Hawaii and Easter Island. Nor did they dominate the Indian Ocean, which was why the Portuguese felt they needed to conquer Goa and Malacca to be able to claim a piece of the thriving maritime trade. Nor were Europeans in control of even the Mediterranean until an alliance of Catholic powers defeated the Ottoman Empire at the Battle of Lepanto in 1571. And I think you would have a very difficult time convincing anyone in China that Europe already dominated the oceans at the time of Zheng He’s voyages.

So what it sounds like you really mean is, why was it European armies that landed on other continents first and conquered the inhabitants?

One factor is certainly geography. As previously mentioned, the Polynesians got as far as Hawaii and Easter Island, but from Easter Island to the mainland is more than 3,500 km with no other habitable islands in between. Easter Island itself only supported around a few thousand people and doesn’t happen to have a large iron mine, so it’s impossible to imagine a huge army of Rapa Nui forging iron weapons and sailing off to conquer the Incas, even if they hadn’t made the mistake of cutting down all the trees. The Bering Strait is much narrower, and people did cross it by land in prehistoric times and settle America, but it would have taken more advanced ships to reach the Americas by sailing east from China than west from Europe, because the distances are greater and the prevailing winds blow the wrong way.

Jared Diamond, in his book Guns, Germs and Steel, argued that there was a good reason: in his theory, the climate and trade zone spanning most of Europe, Asia and North Africa is by far the largest in the world, and also contains all the natural resources it needs, such as copper and tin to make bronze, while the other regions of the world are separated into smaller regions, or were too cold to support a large population, or were too hot and held back by tropical diseases. Then, prehistoric humans killed off most of the megafauna, and most of the domesticated species that were left were within the Europe-Asia-North African zone, spreading within it long before they spread beyond it. Not only animals spread across this region but not beyond it, but ideas, and inventions such as gunpowder.

Diamond argues that most human diseases start out as animal diseases, so this also meant people in this region caught more diseases. This was, counterintuitively, a great advantage: when people from other continents, with many fewer kinds of domesticated animals to give them new diseases and fewer populations sharing diseases with each other, first came into contact with Europeans, the Europeans were carrying vastly more deadly diseases to which the people they met had no immunity, and as a result 90–95% of the Native American population died to disease without European armies needing to fire a shot.

There is definitely some truth to this. The Vikings brought cattle with them to their colony in Greenland, but their herds were unable to thrive there because the climate was too different. The prehistoric relatives of horses went extinct in North America at least 11,000 years ago, there was never anything like it in Australia, and no one ever domesticated the moose or zebra, so no natives of those regions could have had horses or anything like them at the time of contact. Any civilization would have had huge problems when the the vast majority of its population died to disease immediately before foreigners invaded.

This theory, though, does not imply that Europe had any geographic advantage over Asia and North Africa. Diamond suggests that Christianity was especially willing to endorse military expansion, but even a cursory look at the history of other empires suggests that it’s far from unique in that regard. Why didn’t the Mongol Empire, when it controlled East Asia, put the largest and most advanced army in the world on a fleet two centuries before Columbus, arm it with gunpowder weapons from China, and sail across the Pacific to conquer Japan? Well, they did, twice, but their fleets were hit by typhoons and they gave up on that plan. (What would Spain have done if both Pizarro and Cortés’ expeditions had been total losses?) Diamond himself fell back on the same explanation as the other answer: Chinese court politics ended its Age of Discovery, and since that became the policy all over China, its naval tradition died out. When Christopher Columbus couldn’t convince his own small country to fund his idea of sailing west to India instead, he kept trying others until the Queen of Castile said yes. His plan did not make a lot of sense (Even in 1491, educated Europeans knew that the world was larger than he thought and Asia was narrower, so it would not have been a shorter voyage and the ships of the time would not have been able to make it even if America had not been in the way. And many historians now believe that Portugal had already discovered South America by then, but were keeping it a secret so that no one else but them could exploit it.) but all it took was one independent monarch to say yes, and someone in Europe tried the long shot.

Stephen J. Gould’s response to that was to ask: in the world where China colonized America, wouldn’t we be saying, of course it was inevitable? All the European countries were so much smaller and poorer than China and preoccupied fighting each other, or throwing their resources away on pointless crusades. Of course a country like Spain could never have conquered the New World before China got there! Diamond’s response was basically to say that that Gould was focusing on the weakest part of his argument. And you would also need ad hoc explanations for why Morocco didn’t do it instead, or Siam, or Japan.

Unless we someday discover a parallel Earth with a different history, we’re not going to get a chance to re-roll the dice. Portugal was lucky to have a ruler at the start of the 1400s who gambled on the potential of sea power, and then everyone else in Europe heard about it before the rest of the world and raced to catch up. But maybe it could have been a sultan of Morocco, or a shogun of Japan. Or maybe other factors already guaranteed by then that Europe would put together the weapons, ships and belief system that conquered the world before anyone else.

I suggest that the reason was the Mediterranean. European civilizations had lots of places to sail to that could be reached, profitably, with fairly primitive technology. Starting with the Illiad & Odyssey, the Athenian's wooden walls, the Phoenicians, Roman grain ships, &c, then working up to the trade empires of Venice & Genoa. That could all be done within the limited confines of the Mediterranean & Black Seas (with coastal voyages to Britain & northern Europe). So the Europeans had localized maritime trade, and a powerful incentive to maintain & expand it.

Most other civilizations didn't have the advantage of a lot of destinations that could only be reached, profitably, by ships. For China, most sea travel would be limited to simple coastwise voyages. Given early sailing technology, it would be much safer, and more direct, to travel by land. Compare travelling between say Egypt & Rome by sea vs land, vs the same journey between Bejing and Shanghi. The only offshore destination of great interest would have been Japan, and China's attempts to sail there in force didn't work out well :-)

China was a single state. If the emperor decide to stop long distance sea voyages (as did the Hongxi Emperor) they would stop. Europe had the advantage of being split in many states. The ruler of one of these states could not prevent other states from engaging in such voyages.

This idea is developed in Jared Diamond's Guns, Germs, and Steel: The Fates of Human Societies.