Why did so large places of vastly different cultures and histories unite as India?

United Kingdom did not want to grant independence to India. So, as many other cases of revolution, revolting people had to stick together in order to defeat (either by force or diplomacy) the power that has the control. As you state in your example, United States colonies had to join together in order to get independence. Same thing one can say about Spain Empire in South America. Those countries got their independence from Spain fighting together, but later on each of them was an independent country.

In the case of India, one has to see how United Kingdom conquered India in first place. If you see the history, they did not conquered the country in one hit. It was a slow process of defeating each power in India. That division in India was key in their defeat against United Kingdom.

Long before the independence, you can see what happens when only a part of India tried to revolt against United Kingdom, during Indian Rebellion of 1857 what the british did was to sofocate the rebellion using not only their own forces, but also loyal indians as well. Hence, a divided nation can't get independence from a world power.

Another important move is that India had mostly two institutions that were trying to get Independence, one was the Indian National Congress and the other was Muslim League, both of them created long time before the independence. Hence, those two institutions obviously wanted to have control of the whole country (hindu and muslim), not have differents countries. Since United Kingdom allowed the creation of a national Congress, instead of provincial ones, the movement was for a united country.

Since United Kingdom did not want to release India, they tried to create a confederation of provinces, as a way to keep the control or act as a big brother. They had the support from the princes of some of India provinces for that, but nationalistic forces were aware of that move, so they demanded a united country to get complete independence from United Kingdom.

Only when the independence was for sure India was divided, creating a muslim country (Pakistan-Bangladesh) and a hindu country (India). After that, it happens what always happens with new countries, few wars and deportations to create united countries. Here I'll make a guess. Neither Pakistan nor India could divide more, because that will reduce their power to figth each other if necessary. As they actually did several times.

Hence, the lesson is: As long as your enemy is strong, you can't divide yourself.

In spite of the great diversity that may appear to exist in India there are many common threads of history, culture and religion that unify India.

The cultural diversity within India can be compared to a hypothetical nation comprised of say, Italy, France, Spain, Netherlands, Germany, Denmark and Sweden, where each of these countries have become a state within the hypothetical nation. Then each state in that nation will have a separate language and a different culture but the languages will all belong to two major families, Romance languages and Germanic languages. The food preferences, climate, attitudes etc. will all differ among these states, yet it is not inconceivable that such a nation can be formed and will hold together. In fact the EU, although not a nation as yet and has many more countries than the example above, is an agglomeration of such diverse countries functioning as a single entity in many respects.

Unifying linguistic and cultural factors

Similar to the example above underneath the apparent chaotic diversity in languages, food habits, history etc. of the different states of India there are broad but unifying factors. The major differences are not among every state, but between northern and southern India. The states in each region have many common linguistic, cultural and historic factors. So keeping India united is more of keeping two regions united rather than keeping 28 states with 22 languages and hundreds of dialects together.

Most Indian languages fall into two families. The languages of most of northern India belong to the Indo-European family and are descended from Sanskrit. The languages of southern India all belong to the Dravidian family, descended from an much older form of the Tamil language and borrow words from Sanskrit to various extents.

There is also a north-south pattern in food habits. Northern Indian cuisine is mainly based on wheat whereas southern Indian cuisine is based mainly on rice. This is because of differences in climate.

There is a north-south difference in history too that has impacted the culture of the two regions. As most invasions of India (except for European colonization) occurred from the northwest (see here and here for examples), northern India has been subject to more invasions and conquests than the south. So northern Indian languages have a lot of loan words from Persian, Arabic and even Turkish. Such loan words are significantly fewer in southern Indian languages. The traditional music of northern India also has many influences from Persian and Middle Eastern music while the traditional music of southern India retains much of the older styles and forms from before contact with Persian and Middle Eastern music.

Hinduism forms a unifying religious factor across India, north and south. Though there are north-south variations in temple architecture and ceremonies, the fundamental beliefs, mythology and texts are common across India.

The concept of a united India

The concept of the Indian subcontinent being a single entity is quite ancient. Hindu texts have references to names like Jambu Dwipa and Bhaaratha Varsha referring to most of South Asia.

Although the concept of a united subcontinent came closest to political reality during the Mauryan empire nearly 2000 years ago, a united India was not an inconceivable idea to at least the educated classes in India. Movements for greater autonomy and self-determination within the British empire started in the late 1800s. The memory of how the British had conquered most of India by playing one kingdom against another in the disunited political landscape of the 1700s was quite fresh in their minds. When these movements grew into movements for independence, it was clear that a return to the disunited state of the past would not be sustainable and the subcontinent had to be a united political entity to retain its independence. A pragmatic compromise to this idea was made when the irreconcilable differences which had always existed between Hindus and Muslims became very clear and the subcontinent was split into two nations, India and Pakistan when independence was achieved in 1947.

Building united India

Although almost the entire Indian subcontinent (except for a few towns under French and the Portuguese control) was under the control of the British for all practical purposes, the British did not rule all of India directly. Large parts of India were ruled by native kings, Nawabs, Nizam etc. who had a limited autonomy under British supervision. So when the British granted independence to the areas they ruled directly, the status of these princely states was not fully settled.

The difficult task of integrating the princely states into united India was directed by an Indian leader, Vallabhbhai Patel. The rulers of these princely states were offered incentives like privy purses to allow them to retain their luxurious lifestyles. Most of these kings and princes willingly integrated into India. There were a few holdouts like Hyderabad, which required military action by the newly independent India.

The towns under French rule were integrated into India peacefully in the mid 1950s and those under Portuguese rule by a military action by India in the early 1960s.

Staying united

The unifying factors described above are one reason for India remaining together. There have been secessionist movements like the Khalistan movement in the 1980s and movements in some north eastern states, but those movements have been mostly defeated by counter-insurgency action by the military. An improving economy is another incentive for remaining a united country as many regions of India do not have sufficient natural resources to be able to maintain an economy even at the current level as separate countries.

Although several states were split into smaller states in the last 15 years following demands that have existed for decades, none of then seceded from India.

Thus in spite of the chaotic diversity that India may present on the surface, there are unifying patterns in Indian languages, culture, cuisine and history, with differences between northern and southern India. These broad differences are bridged by religion, lessons learnt from history to avoid disunity and economic need to remain together.