You see a problem with being MORE specific than simply European? White Americans identify themselves ethnically only to be more specific than the obvious.
Usually white americans get more specific about their backgrounds, Irish, polish, german, etc. whereas blacks cant because there is no trail to their roots because WHITE men kept them uneducated and in barbaric conditions for hundreds of years. As for hispanics, there hasnt been quite tribal warfare and explosions of world wars which keep them as divided and distinct as say the French and Germans and ENglish and Italians, so they dont get so specific about their backgrounds.
So, based on that, most white americans refer to themselves as a german, frenchman, english, aussie, etc, etc. When the term of white person does come up, it is usually gonna bring along with it some racist baggage. Thats the only time they use it.