As an American I have no real way of knowing rather Europeans have a better sense of cultural awareness compared to Americans. I know that Americans are generally socially unaware or sometimes lack the openness to really understand others points of view, even if they fit into the social norms, so I'm hoping we just aren't up to par in that area and maybe Europeans are.
No, Europeans don't have a better sense of cultural awareness than Americans. Europeans are generally culturally aware of an area about 1/3 the size of the United States. They think that because that area includes different nations, that it makes them culturally superior. Americans know 1/3 of a vast continent -- their own -- pretty well. So there.
No, Europeans do not have a better sense of cultural awareness than Americans do, because culture is not necessarily better just because it is European. European culture is interesting, but American culture has so much influence on the rest of the world. Americans are some of the most culturally aware and generous people on earth.
Americans and Europeans have different cultures, but I do not believe that either is more aware of their culture or values their culture more than the other. Many European cultures are more well-defined, since America is a blend of many cultures, but Americans have just as much appreciation and respect for their blended culture as their European counterparts do.
America has had different cultures coming in for a very long time, and so have the European countries. They are not any more aware of different cultures than we are, and have the same amount of conflict from different races coming in that we had through out our history as a nation.