For the record... As much as the Spanish-speaking cultures cry about it, they are not "Americans". You never hear Canadians and Brazilians fighting to be called Americans (the Canadians are likely to kill you if you try).
American is the diminutive of the United States of America just like Mexican is the diminutive of Los Estados Unidos Méxicanos.
And I hate "Estadounidense". If we're Estadounidenses, so are Méxicanos.
And there is no continent called America. There are North America and South America, which are the Americas, but never America. The only America is the U.S.A. It's the same with how the United Mexican States call themselves Mexico for short.
United States is the equivalent of Bundesrepublik or Estados Unidos. You don't hear Deutschen calling themselves Bundesrepublikaner and you don't see Méxicanos calling themselves Estadounidenses. Same thing. We are Americans and the short form of our country is America.