bad. Most of the US airlines that appear to have decent management are LCCs that wouldn't want to go to Europe. A couple of the legacies could probably survive, but I think that European airlines have better management because it's more difficult to make money by driving a business into the ground over there.
It also depends on if the European airlines would hire American pilots for their routes and vise-versa. If not, then the value of currency could play a large role. Can you imagine Air France trying to make JFK-LAX profitable by paying their pilots in Euros, while American can pay their pilots half as much since they're being payed in dollars?
There's a lot of factors that go into this, but I do think it will happen eventually. It's just part of globalization.
I can't imagine, however, getting on with a major and having them tell me I'll be based in Brussels or some crappy place like that (I hated it there)