They dont consider themselves so.
I lived in Brazil and never once heard gringo used in this way. It always meant (United States of) American.
Surely Americans can decide for themselves what they call themselves without outsiders deciding what to call them.
Sent from my SM-G955U using Tapatalk
Never have I ever heard of this.
Indeed, this turns the stereotype about Canadians on its head.
There are also a small number of Canadians who believe the Earth is flat.
Actually we do have, or did have, a Canadian poster who kept typing “USAian.” You do run into people like that now and then.
As others have pointed out, for a variety of reasons, “American” simply makes sense as the demonym for people and things from the USA, and does not make sense for people and things from Canada, since in Canadian English there is no continent called “America.” In Canadian (and American) English, Canada is located on the continent of North America. (It is worth noting that in other languages these semantic nuances are not always the case.)
It is further worth noting that he demonym “American” has been applied to people from what is now the United States since before there was a United States. “American” was first used in 1568 and meant the native inhabitants of the New World, but by the mid 17th century the term “English-America” was being used to describe residents of the 13 Colonies, and that soon shortened to “American”
Bear in mind that at the time, most of what is now Canada was not under British control; it was mostly a French colony, so a person talking about “Americans” was almost certainly not including people in Canada. Most of Canada was not conquered by Britain until the Seven Years War, at which point the demonym “American” was pretty well established as meaning the 13 Colonies and parts west (Acadia - today’s Nova Scotia and part of New Brunswick - fell to Britain in 1714, but were almost wholly populated by French colonists.) So the distinct word “Canadian” - derived from the French “Canadien,” because if you think about it, had Canada been English the whole time, the word would be “Canadan” - was therefore needed as a distinguishing demonym, and became more of less official when Upper and Lower Canada were created in 1791.
The term “American” actually arose first in BRITAIN, not in the colonies; colonists early on generally would have called themselves British (or English or Scottish) and, later, “Virginian,” “Pennsylvanian,” etc.
I’m a Canadian, and in real life I’ve never met a single person who objects to the use of the term “American” to mean someone from the USA.
They may also use “Yank” (as a Canadian is a “Canuck”).
I can believe that some people on a chat site somewhere object to it … but they are, believe me, not representative of any substantial minority that actually exists up here. 
Every other country in the Americas has a well established word for its residents that no one fusses about: Mexican, Honduran, Brazilian, etc.
The U.S. has a well established word for its residents – that preceded all the others, by the way – that no one fussed about until recently: American.
I don’t know who figured that North America and South America comprise one continent, America, but I’d love go get their ear and tell them how wrong they are. I suspect this bit idiocy fueled people thinking “Hey, I live on the continent ‘America,’ so I’m an American, and those people from the U.S. are usurping my adjective!”
Anyway, folks who get their panties in a bunch over it are welcome to spew their nonsense, but as a U.S. citizen I’ll call myself an American, and there’s nothing they can do about it. 
Texas, USA.
Admittedly no encounter with such a person in person, but YouTube comments or maybe Cracked.com comment, cannot recall which. Of course, there are ALL kinds of people on the Internet, though. At any rate, must be a tiny tiny number of Canadians.
But then, when the Monroe Doctrine states “America for the Americans”, what does “America” means?, it seems to me that the definition of “America” is not so clear cut.
Cat Whisperer, on this board, had made this argument.
A Google search of the most trivial effort will find many, many people who are really offended by the word “American.”
It was rather less clear cut in 1823, when it was not entirely clear what territory the United States would end up occupying.
I guess I don’t move in easily-offended circles, then.
I suppose Tierra del Fuego was in the sights of some wild eyed American imperialst? 
I think there are 2 definitions of “America” (at least) one is “The USA”, but the other is “All these landmasses with ‘America’ in the name”. Here we generally use definition number 2, (the USA is “Estados Unidos” when we are feeling kind and starts with “Yanquilandia” and goes exponentially worse from there when we are not).
So when we think about y’all using “American” to mean only people from that part of definition #2 America it certainly rankles.
Personally I understand both sides, you have called yourselves “Americans” all your lives, you are not going to change it because it hurts the feelings of people from other nations that use the term differently.
But you should, if you want to avoid the charges of cultural insensitivity often leveled against (def #1) Americans, at least understand why it bothers people.
For decades, if not centuries, the term for “All these landmasses with ‘America’ in the name” was (and still is) “The Americas,” not “America.” I’m sorry someone went and decided to confuse the issue by appropriating “America” and “American” and giving them new definitions. We had perfectly good words that no one objected to and then some fool decides to change the rules and start blaming us. Now THAT is what strikes me as culturally insensitive, and yes, it bothers the hell out of me.
These are not NEW definitions. As shown by the Monroe quote, the name “America” has been used to name all the lands discovered by Columbus (and mistakenly assumed to be discovered by Amerigo Vespuccio) for centuries.
And as I said, I understand why it would bother you (as in def #1 Americans) to suddenly being asked to change the definition you have been used to all your life.
All I said is, consider that it bothers other people the other way too, have it in mind, nothing else.
There is some disputed evidence that Vespucci landed on the American* continent* before Columbus did. Columbus didnt land on the Continent until 1498, and still didnt think it was a new continent. Vespucci had a earlier (and disputed) voyage which landed in 1497 and definitely landed and showed that South America was a continent in 1499.
Definitions again!, the Caribbean islands where Columbus landed in 1492 belong to (Spanish definition) America.
I’ve occasionally heard reference to ‘Carolina’ without the ‘the’. Also not often but occasionally a Dakota reference without the ‘the’. But unusual usages like that would be a thin reed on which to build an argument about people being justified to be ‘offended’ by the much more common versions.
Nobody can give a good practical argument why Americans should worry about calling themselves such, especially in English. ‘Because we feel it’ doesn’t cut it. Americans could as easily tell Latin Americans not to call us ‘Norteamericano’ in Spanish, but rather ‘Americano’ (without even getting into ‘gringo’, which is a slur).
An ambiguous quote/slogan by Monroe a long time ago aside, calling the people of the United States of America ‘Americans’ is longstanding standard English language usage. It’s logical. There is no sensible alternative. ‘North American’ in English means the American and Canadian people/society/culture (or in case of NAFTA the US, Canada and Mexico in terms of their governments, though never AFAIK are the Mexican people, society or culture referred to as ‘North American’). The people offended by it just need to live with it I think, as they did for the long time it was standard usage and nobody complained.
Lets ask the Dutch
A trivial search will find many, many people who are really offended by the words straight*, black, and white, but no one else’s much worried about not using those, either.
*I had an RA who would throw a fit when people used the word straight to indicate direction. It made driving with her an ordeal.
It seems to me if you want to talk about chauvinism, saying Europe is a different continent from Asia, but North and South America are not has got lots of other things beat.
And what about Antarctica?