Let’s get a few things straight here
North America is a continent. South America is a continent. (Central America is like some sort of weird connective tissue of a landmass, I guess.) The Americas are not, nor ever have they been, a continent en masse. The English and Spanish languages follow roughly the same division as the continents, with some exceptions: Spanish creeping it’s way north of the Isthmus of Panama, and Portuguese making it’s dichotomy-ruining foray to the south.
In Latin American Spanish (I don’t know about Spanish Spanish), someone from, say, the United States is a norteamericano. Rightly so. Spanish is spoken widely in South America, and North America’s asymmetric political presence in the hemisphere necessitates the distinction. Saying you’re American isn’t specific enough.
But in American English (I can’t speak for, or speak, British English), American is synonymous with United States-an (I made that up; did you notice?). It’s a legitimate abbreviation of the country’s official name.
I get that when I’m in, for instance, Nicaragua, I have to adjust my customary manner of speaking to say not that I am American, but that I am from the United States. And if I am speaking Spanish, I’ll say that I’m North American. That’s all fine and dandy.
But I’ve had pseudo-sophisticated norteamericanos call me out before when, speaking English with other United States-ans (see? there’s no way to make the distinction without awkwardness in English), I said that I would be leaving America, even though I wouldn’t be leaving the Western Hemisphere.
And that’s just annoying (he knew what I meant).
The correction is equivocating on a cognate. The distinction is linguistic and cultural, not universal. I’m from America. I left America to come to Nicaragua. If I were writing this in Spanish (let’s get real; my Spanish isn’t that good yet) I would say it differently. If you are from South or Central America and I make the mistake in conversation with you, then please correct me. If neither one of those is the case, kindly keep your appropriated indignance to yourself.