I don't recall him having any influence at all. People everywhere have enough common sense to discern that communism is an evil and failed economic system. Private capital and ownership are inherent rights of man, given by God not by man or government. You don't need a PhD for that.
Yes, communism would have fallen in Eastern Europe and the Soviet Union without the influence of Pope John Paul II, because there were other forces in place in the Soviet Union to cause the society to collapse. Pope John Paul II was certainly instrumental, but he was not the sole cause of the change.
While the Catholic Church and the Pope held sway over Europe during the Roman Empire and even a time after I do not believe they've held that same amount of power since and rightfully so. I believe communism would have met its end in Eastern Europe and the Soviet Union regardless of Pope John Paul II's position.
The reason communism never works is because it is never really communism; rather, it is dictators who are hoarding almost everything and then doling out even what remains to their people, who are being treated like peasants. And even with the citizens, it's not all equal. There are still haves and have-nots. Systems like this will always fail.
Communism failed in Eastern Europe because it was not a viable economic model. The USSR could not keep up with the military spending of the west. Their technology was being outclassed and the people were being impoverished. While Pope John Paul helped quicken the fall of Communism in Eastern Europe, its fall was inevitable.
Yes, it still would have fell, because of the wars that were fought. Countries like the US for example have fought very hard to keep communism from spreading, and have fought to get rid of it. The pope did not have all that big of an impact on getting rid of it.