Did Russian winter seriously contribute to German defeat on the Eastern Front in WWII?

Weather? Not so much. Russians? Absolutely.

Simple fact:

From the time of the German invasion to the battle of Stalingrad the Russian/soviet forces NEVER faced less that 100 to 110 German divisions. The "allies" faced the maximum of 15 at the Battle of the Bulge. The simple fact of the matter is that Stalin's Soviet Russia defeated Germany. The rest of the "allies" were just along for the ride.

Yes and No.

Yes.. Winter totally defeated the Nazis. Had the summer of 1941 been 12 months long, Germany would have handily won!

No. It is Russia. If you invade Russia, you have to take Winter into consideration.

Whilst many of the other points are excellent, also bear in mind that there are times known as the Rasput**sa or "the time without roads". Whilst many of the tanks would have coped, remember that not every vehicle in a militiary convoy will have tracks. Many will be wheeled vehicles, which will struggle with the conditions.

Logistics is the lifeblood of any army, and together with poor equipment and overstretch, the added difficulties of dealing with deep clay mud combined with summer gear is going to make any army miserable. Miserable soldiers fight less effectively.

The conditions during that particular Russian winter were a contributing factor (as has been discussed in the other answers). However, there were other significant contributing factors to the German defeat.

One of the biggest factors beyond the weather conditions was actually the tactics used by the Germans. Blitzkreig tactics work great, until you end up advancing faster than you can increase the size of your fighting force. The number of troops you need to keep using such tactics increases as you claim more territory, and the large open terrain of the Russian plains made the required number of troops go up exponentially. This, combined with the fact that the Russians largely gave up territory with minimal casualties early on meant that the German forces were spread rather thin when the Red Army really started with the counteroffensive, and thus had issues effectiviely fighting back. This was significantly compounded by the winter itself however, so it's hard to say what the primary cause of the Russian victory was.

In short though, the whole operation is a great reminder of the fact that invading.a country with a significantly larger potential fighting force than yours without proper preparation is a remarkably effective way to lose a war.

The Russian winter was a contributing factor. The German winter is not like the Russian plains winter where the German army advanced. The German army was reportedly still wearing summer uniforms when the Russian winter hit. The winter was one of the coldest for that time period. Hitler refused to send winter uniforms when initially requested. Gasoline won't ignite at temperatures lower than -42 Celsius and diesel is hard to start at low temperatures. German armament had difficulty with the cold as well.

Part of the Third Reich's military problem was its devastating quick early victories.

Germany invaded France rapidly alongside a ~900 km front line, adopting the WWI strategy to enter via Belgium through the Ardennes surpassing the Marginot defense line and catching France by surprise as those considered passing the mountain range with tanks impossible. This gave a huge confidence boost to the military brass and let to overestimation of their forces' capabilities. When invading Russia, their forces were hardly better equipped or in larger number, however the eastern front line was ~2400 km.

The battle of Moscow

The battle of Moscow was the turning point for the war. In the previous battle of Smolensk in early September 1941, German troops had closed in on and defeated Russian forces.

Hitler already ordered the forces to divide in late July 1941, because he wanted to conquer Leningrad and the economically important regions in the Ukraine, so he sent a tank brigade (Panzergruppe 2 and 3) to take part in the battles of Kiev and one in Leningrad, which they did after the battle of Smolensk. He also wanted to capture the Crimean peninsula in order to eliminate the threat of Soviet aerial attacks on important oil fields in Romania.

Supply problems

Those two battles took longer than expected, however. Thus the reinforcement for the front in Moscow to which the tank brigades were supposed to return delayed. Additionally many tanks were out of use and in repair in Germany so out of the remaining tanks only about half were ready for use. Also only about a quarter of motor vehicles were ready to be used, and even those were only provisionally repaired. Additionally, supply bases for army groups middle and south (Heeresgruppe Mitte/Süd) ran short on fuel, so only the supply base for army group north (Heeresgruppe Nord) had enough left because rails were better developed and they could be refilled via ports in the northern sea.

In order to secure supply, rebuilding infrastructure in Soviet territory was necessary, even setting up a new one as there were almost no paved roads and the Russian rails had a wider gauge, which did not fit for German trains and had to be modified time consumingly. There were also not enough supply trains available and Soviet partisans attacked some of them.

Thus Germany was not able to compensate the losses from previous battles.

Early weather trouble

Initially, in August, German army command rendered Moscow impregnable before winter, but a disastrous defeat for the Red Army in Kiev let Hitler redecide and order the conquest of Moscow before winter.

After the attack for Moscow started in early October German troops decided some early battles for their side (Oryol, Bryansk, and Vyazma) and crossed the Volga river. In mid October the mud season started with heavy rainfalls rendering the land impassable and a German supply drop from 900 t of material per day to only 20 t until frost settled in early November. It still took another two weeks for supply materials finally arriving at the front line, and when biting frost settled on the 6th of November already, German troops were still without winter clothing.

In the meantime the Soviets gathered their troops, strengthened defense lines and prepared for a counter charge.

Late weather trouble

When German troops continued the attack they encountered heavy resistance and failed to achieve aerial superiority due to some of their planes having been relocated to the Mediterranean. Despite the resistance, German troops marched on and won several battles (Yepifan, Dedilovo, Stalinogorsk, Mikhaylov, passing of the Don river, and Skopin) until field marshal Fedor von Bock informed the army command of the troops' exhaustion. Army group middle (Heeresgruppe Mitte) received response to continue the battle with a last exertion as they assumed the battle was fought between the last Batallion on both sides.

Temperatures fell below -35 °C in late November and the German troops suffered heavy losses due to freezing while Soviet troops were fully equipped with warm clothes. Despite that, the last German forces took the town of Krasnaya Polyana and one army patrol even reached the Moscow suburb Khimki about 8 km from Moscow.

When the Soviets started their counter attack in early December already 21 of the 34 Eastern Siberian units had arrived in Moscow.

East Asian situation

After the Russo-Japanese war in 1904 Japan increased influence in the Chinese region Manchuria, creating a vassal state, Manchukuo, in 1933. They were unable to settle for a border agreement with Russia causing some tension between the two parties. A small group of Mongolian troops sat down in what they considered their side of the border for their horses to grass, but they were ran off by Manchukuo forces. A stronger Mongolian force returning later let the Manchukuo call the Japanese for reinforcements, but the area was inapt for battle as neither side had any proper railway system, paved road or train station close to the region.

Upon the conflict's growth the Russians set up a railway system on their side so they were able to relocate a large amount of troops, artillery, ammunition and even tanks below the radar of the Japanese. When they attacked to drive out the Russians they were surprised by the strength of the enemy's troops and suffered a devastating defeat.

This led to the Japanese deeming the region as too unimportant to fight in large-scale battles for it so they signed a non-aggressive pact with Russia turning their full attention south towards China.

Stalin mistrusted Japan and kept his troops stationed in East Siberia until Russian agent Dr. Richard Sorge, disguised as German newspaper correspondent (Frankfurter Zeitung) transmitted to Moscow in mid August that the Japanese leadership decided to finally lay down any offensive plans to attack Russia. Stalin then started to send the troops to the western front.

Hitler expected Japan to join the attack on Russia according to the Tripartite Pact, but against his vision his ally abode by the non-aggressive pact with Russia and focused on the war with China and later the Pacific war.

Stalingrad?

I am leaving out Stalingrad now as that was actually only one point in the continuing defeat of Nazi Germany, albeit the most devastating one, but the problems of German troops have all showed up months before already.

But here is the order Hitler gave to the German troops in mid December on how to battle the stronger Russian forces:

An die Heeresgruppe Mitte

1. Der Führer hat befohlen:

„Größere Ausweichbewegungen können nicht durchgeführt werden. Sie führen zum völligen Verlust von schweren Waffen und Gerät. Unter persönlichem Einsatz der Befehlshaber, Kommandeure und Offiziere ist die Truppe zum fanatischen Widerstand in ihren Stellungen zu zwingen, ohne Rücksicht auf durchgebrochenen Feind in Flanke und Rücken. Nur durch eine derartige Kampfführung ist der Zeitgewinn zu erzielen, der notwendig ist, um die Verstärkungen aus der Heimat und dem Westen heranzuführen, die ich befohlen habe. Erst wenn Reserven in rückwärtigen Sehnenstellungen eingetroffen sind, kann daran gedacht werden, sich in diese Stellungen abzusetzen.“

2. ...

This translates to:

To the army group Middle:

1. The Führer commanded:

"Larger evasive maneuvres cannot be executed. They lead to a total loss of heavy weapons and devices. Under personal commitment of the persons in command, commanders and officers the troops are to be forced in position executing fanatic resistance disregarding any enemy breakthrough in the flanks or the back. This type of warfare is the only means to achieve the gain in time necessary to bring in the reinforcement from the homeland and the west, which I ordered. Unless those resources have arrived in rearward positions a retreat to those positions cannot even be thought about."

2. ...

I am unsure as to whether he actually ordered the reinforcements. He also ordered the desert corps to fight fanatically to the last man rather than to give up or retreat.

Summary

So to sum it up, disadvantageous weather conditions, underestimation of the enemy, overestimation of the own troops' capabilities, supply problems, unexpected ally behavior all played part in the German defeat. Though the weather may have been a key component as it weakened the Wehrmacht severely.

Other answers have dealt with the main things but I think you are missing one important point.

Even assuming that Berlin and Moscow are equally cold there is a huge difference between holding your own territory (defensive operations within Germany) and maintaining long supply lines through hostile territory in bad weather. If the German army had a constant supply of replacement equipment, cold weather gear and hot food things would have been easier for them.

There is a good reason why ancient armies did not operate during winter. It is a lot easier for a defender to hold a position in winter than it is for an attacker to take it. Even assuming that both armies were from the same climate and equally capable of winter operations.

Average temperatures are irrelevant, unless you plan to repeat the battle every year. What matters is the temperature at the time. To quote Wikipedia:

The European Winter of 1941-1942 was the coldest of the twentieth century. On 30 November, von Bock reported to Berlin that the temperature was – 45 °C (–49 °F). General Erhard Raus, commander of the 6th Panzer Division, kept track of the daily mean temperature in his war diary. It shows a suddenly much colder period during 4–7 December: from –36 to –38 °C (–37 to –38 °F). Other temperature reports varied widely. Zhukov said that November's freezing weather stayed around –7 to –10 °C (+19 to +14 °F) Official Soviet Meteorological Service records show the lowest December temperature reached –28.8 °C (–20 °F).

That winter was quite cold in Germany as well, but by that time it was too late to supply the army which was some 2000km away.