The funny thing about it, is that at the time the name "United States of America" came to be, "America" was the full continent and the name of the new nation simply reflected the fact that it was a union of states in that continent. I think the name "America" for the country got stronger because people from there were calling them selves "Americans" for a lack of better demonym.
Actually, in Portuguese and Spanish there is the word "estadounidense" that is basically "Unitedstatizen". But this is a formal word, used only in books (and Wikipedia). At least in Brazil, we use "americano" in the day by day for both someone born in the USA or in the continent (which is still called "America", no plural).
As a Brazilian from Sao Paulo, I used to consider myself white. But now I live in North America, and I don't think that anymore.
If you think about, many very white people in Brazil get nicknamed "Alemão" (German), independently of their origins. That's a sign that we are not so white after all.
The white/non-white dichotomy is purely an American construct. When they say "white" they mean "anglo-saxon". In my opinion anglo-saxons are actually red, not white, but that's a differnt argument.
The rest of the world uses ethnicity/descent to describe skin colour/appearance.
