Actually America refers to the entire Western Hemisphere; i.e. the Americas, so technically we are all Americans; Canadians, Peruvians, Mexicans, Jamaicans, etc.
I don't know why you guys had to call yourselves Americans, or Untied States of
America, in the first place since the term is supposed to refer to everbody on this side of the Atlantic. That makes as much sense as only calling a person from the UAE, United Arab Emirates, an Arab; the term applies to all of Arabia. If you catch my drift.
Ah well, you guys are called Americans now, whatever. Most Canadians though never say America, we say the States or the US, I guess out of principle or just to annoy you guys.