I guess that’s just too bad for your whiny Canadian friends. If you go anywhere in the world and say you are an American, 90% of the time people will assume you are from the U S A
Which is a different statement (as has been noted already in this thread) from saying that you are from America.
My memory of listening to Canadian radio shows and talking with employees at the Canadian division of a company where I worked was that they generally used American as the normal reference to a citizen of the U.S. However, they were more likely to say United States, U.S., or the States when refering to the country–and occasionally bristled at the use of America to identify the nation to the South (or to the North when they were in Windsor).
I watch a fair amount of BBCA (BBC America) and notice that Brits use ‘America’ to describe the US more than we do in the US.
BTW does BBC America broadcast in Canada? 
This bit about “America” being offensive to Canadians is news to me. Be aware that there are some who will be offended at you no matter what nomenclature you use.
I do agree that we will normally refer to “the US” or “the States”, but “America” is by no means unknown. If I think about it really carefully, I will agree that “America” has a slightly jingoistic overtone to it (perhaps through popular songs? Since the first draft of this post, I’ve had that “They come to America” lyric running through my head … though that’s the only line of the song I know and I can’t remember who sang it) and some might well pretend to be offended. But really, most reasonable people will simply use whatever term seems to fit best into a sentence … and “the US” or “the States” is normally preferred here, just as a matter of style and habit.
I have a British uncle who asked me (when I was about 15 or so) whether it ever bothered Canadians when people called us Americans (meaning the continent - not by mistake). I was and still am rather puzzled by the comment - nobody has ever referred to me as an American (except by mistake when I was in Russia). Perhaps his question was related to English bristling at the thought of being called European?
However, this is the age of self-actualization by taking offense, so I will simply note that I am offended at the thought that somebody might think I would be offended by how Americans choose to refer to their country. Send money.
Oh yes, and if I should want to satirize an American, I would definately sprinkle the performance liberally with references to America, preferrably pronounced “uh-MAYR-ka” - emphasizing the difference, as in all caricatures.
No, we get BBC Canada, of course. (Also BBC World News and BBC Kids…)
It was Niel Diamond’s Coming to America…which of course does not have the same ring as Coming to the United States of America.