This really is one of the stupidest debates I've seen in a long time.
The USA is the United States Of America. They are states of the American continent that chose to unite. That is LITERALLY what it means. It's not just an arbitrary name.
It's the same exact deal as if people got butthurt about being called "Norwegian" instead of "European" because Norway's not a part of the EU. The EU is the European Union. A Union of European states. The common usage for the parlance isn't nearly the same for this to be a perfect example but, fundamentally, that's exactly how stupid this argument is.
Calling Americans "Statesians", "Staters" or any of the other names I've seen suggested in this thread are purely moronic. "Statesian" is a completely meaningless demonym. It could just as easily apply to ANY state ANYWHERE. It does nothing to narrow it down to meaning citizens of the United States.
If you call a Texan a Yankee, you better be prepared for the good possibility of finding yourself with a face full of knuckles. Same goes for using the term basically anywhere south of the Mason-Dixon line. A Yankee is a Northerner. Southerners (or, to use what would be the equivalent term for a Southerner, "Confederates") have a fair tendency to get pissed off at being called Yankees. It's sort of a matter of immortal Southern pride. Calling a Southerner a Yankee is basically like calling them a wuss.
For Texans it's a bit different. We're technically Southerners, but the whole Southern pride here is a peripheral issue to pure Texas pride. We used to be our own country. We're Texans first and second, Southerners/Americans third. Maybe it's not readily apparent to people elsewhere, but while the South has it's whole "The South Will Rise Again" notion of pride that the South could and should stand on its own separate from the USA, Texas goes beyond that since it is fully secure in the knowledge that Texas is the only State genuinely capable of surviving completely on its own. We've done it before and we can do it again. We're even the only state with our own military force, and the only state allowed to fly its State Flag at the same height as the American Flag. It's not hard to find Texans who will tell you they're from "The Republic Of Texas".
Honestly you're MORE likely to get punched in the mouth in Southern States other than Texas. They take it more as a direct insult to their primary point of patriotism.
Obviously, levels of patriotism vary wildly and a lot of people don't actually give a damn, but you DO run a fair risk of finding you've seriously pissed some people off.
And along this same basic train of logic, why the hell would Canadians or Mexicans WANT to be called "Americans" if they're supposedly so offended by the place? Shouldn't they, likewise, be proud to be called what they are (e.g. Canadians and Mexicans)?
And on yet ANOTHER level, "America" is a catch-all term for formerly independent political entities on the North American continent that opted to join the Union. Mexico did not. If they join they can be "Americans". Same goes for Canada and Canadians. Since they obviously have no interest in joining, they've obviously opted to have specific names for their countries that are not mostly generic to the continent on which the union of states in question is located.
Mexicans and Canadians don't have any NEED to be called "Americans".
Indeed, this entire thread is further frivolous in the fact that the continent IS NOT ACTUALLY CALLED "AMERICA". No, this continent is called "NORTH AMERICA", and no one, anywhere, debates that Mexicans and Canadians are "North Americans". That's just understood, because THAT is logical.
So yea, by all LOGICAL thought, the only relatively simple demonym that should be applied to the citizens of the United States Of America is "American". And if, for some stupid reason, you feel that this fact still offends you, your next best option is referring to them by their State-specific demonym (e.g. "Texan", "New Yorker", "Floridian", "Mississippian", "Hoosier", "Nutmegger", "Wisconsinite", etc, etc, etc), but good luck remembering them all.