The cold war is certainly a factor, as is the quick economic recovery of (West) Germany. I have rather big doubts, though, that US occupation caused West Germany (and France) to become a champion of European integration.
I also don’t think that Europe has yet to handle a serious problem. The fall of the Soviet Union is kind of the biggie here. And, sure, the cold war played a large role during the recovery of western Europe after WWII, but, looking back at the disastrous recovery of Europe after WWI and comparing it to the stellar recovery after WWII tells me that this is also quite a big problem which Europe sort of solved. With help, but help can only get you so far.
Well, sure, the Cold War made a European war (as in: a war between West European nations) unthinkable but it is my very strong suspicion that any European war became much less likely independently of the cold war.
There have been long periods of fragile peace in Europe (one example would be the more than forty years of peace between Germany and France between 1871 and 1914). I don’t think the current European peace is anything like that. The events which led up to WWI were by no means huge problems. A little will, a bit of diplomatic elbow grease, eminently solvable if you really wanted to.
The violent disintegration of Yugoslavia in the 90s did not cause huge conflicts in Western Europe. It even looked as though Western Europe didn’t care all that much for most of the time. A conflict between the relatively unimportant Austria and tiny Serbia – a century ago, also on the Balkans – could, however, start the chain reaction which resulted in WWI. The peace was so fragile that it couldn’t handle a local conflict.
The US did indeed encourage European integration but there is always the question as to whether it works. The US always wanted to bring Europe on a course which would lead it to a peaceful future – you know, so they don’t have to come over every other year and fix everything. That didn’t work so well after WWI and a disillusioned US left Europe.
Umm, the credit goes to US occupation of Germany and tension with the USSR.
Europe has yet to handle a reasonable problem on its own.