My first instinct is
but in all fairness I like very few of my countrymen.There is no continent called “America.” There’s a North America and a South America.
America is our last name. It’s a good one. Mexico goes by it’s middle name because that’s better than it’s first or last name. Back when there was a USSR all four of their names sucked so they just used initials. The English can’t even figure out what their name is. Canada only has one name so they have to use that.
Anyone else ever wonder how someone from the US Old South (former Confederacy) who still worships the Confederate Battle Flag feels when he’s in Australia and is called a Yank?
Jeff: this is an example of the graduated meanings of “Yankee”: to a foreigner, a Yankee is an American. To a (southern) American, a Yankee is someone who lives north of the Mason-Dixon Line. North of the Mason-Dixon Line, a Yankee is someone from New England. In New England, Yankees are from Vermont, and in Vermont a Yankee is someone who eats fruit pie for breakfast and doesn’t speak French.
And many sports fans in various countries will always regard a Yankee as a New York baseball player. 
Plus, as people routinely point out in these conversations, Brazil and Mexico are among the other countries which are called United States of (Something).
And nowadays it has come to mean either “from the US (no offense intended)” or “outsider from anywhere who’s being a jerk about things not being like back home (it’s not an insult if it’s true)”, and you really need to think about the context to know which one is meant.
As an aside, in Cockney Rhyming Slang an American is called a “septic”. As in “septic tank” = Yank. If you are anti-American you are a “listerine” since Listerine is a brand of antiseptic mouth wash.
I think the OP’s got it wrong: What some Canadians object to is “North American” being used to describe US citizens.
In German, an American is typically called “ein Amerikaner” (and btw. that is also what we call our variation of a black-and-white-cookie). When needed for clarification we might say “US-Amerikaner” (using the German pronunciation of the letters U and S: “oo es”). In everyday speech you can often hear the short form “ein Ami” (pronounced like “Emmy” but with an A at the beginning). And sometimes you can hear people using Yankee or even G.I. (both pronounced the English way).
Btw. When is was in school in the 80s and 90s we learned that there are five continents: Europe, Asia, Africa, Australia and America.
In Spain we called them “continents” when we were little, “parts of the world” once we’d had the explanation about Oceanía not being a landmass so much as a sprinkling of landmasses (we got Oceanía instead of Australia). There were also versions which used Antártida as a sixth part but only for physical geography; when talking about human geography, “that biggish landmass with no permanent inhabitants” didn’t count.
Actually, that is very similar to what we learned in school in Germany.
Another German word for “Kontinent” is “Erdteil” which is literally “part of the world”. Same thing about Antarctica. The term Oceania used to be rather uncommon - I don’t think we used that in elementary school.
Australians solve the problem by always asking me first if I’m Canadian. If I’m not Canadian, then I can be American. Australians are very sensitive about naming things correctly, and expect Canadians to be the same. I’ve not met enough Canadians to have an opinion about that…
I occasionally use “Yank”. But I don’t see a problem with American, really.
(Cue *recent *South Park scene where the PC boys are just discovering each other.) It’s relevant, but I can’t link from the office.
See this letter in Le Devoir, from a few weeks ago.
The word Étasunien is used occasionally by most French-language newspapers in Québec (although I haven’t seen any of them ban Américain from use):
Example 1
Example 2 (this one uses afro-étasunien where one would expect “African-American” !)
Example 3 (this one mixes américain and étasunien).
Example 4 (from a politician acting as a columnist)
Example 5
Hey, I even found some resentment in the comments from an article in the Globe and Mail (scroll down to “Jimpse” 's comment and replies thereto).
ETA: Example 2 above should have used this link.
Only, in Spanish it would be Chinese. El tipo ése habla en chino.
OK, I can see an argument for calling Europe and Asia separate continents. I can see an argument for calling the Americas all one continent. But how in the world do you justify splitting Eurasia while at the same time lumping together North and South America?
“Historical reasons”, of course.
‘The Americas’ works well as a combining term for the North and South American continents. ‘Eurasia’ works well for the big one with no clear division. The funny specs for continents and oceans don’t create much of a problem in life, but they create a lot of threads.