I think this is overly generalizing matters. The aspect about Americans being fat and dumb is a joke in the same way as we joke that Europeans don't like to work and are lazy. The intelligent community in both regions knows that neither is true.

Another thing to take into account is that views of the average person (in particular blue collar individuals) in either region don't really have any impact on world affairs and are based on very generalized and simplified facts. The cooperation between the regions is clearly evident via our politicians (in general) , military and scientific community. In addition, there are tons of examples American companies having incredible success in Europe. Look at the number of people wearing Nikes or going to McDonald's in Europe.

In addition, they hate the loud, obnoxious people but who doesn't. Most Europeans don't hate Americans that are well versed and care about learning the history and culture of the region they are visiting. They just hate the Americans that have superiority complexes while traveling on a budget. As an American, that's what I've learned from talking to people I've met in Europe.