Well, "Asian" or "European" or "African" are not languages either. Why would "American" be? ;)
The only measure of correctness for language is use, and when speaking in English, American is only commonly used in one way--in English, American is almost universally unambiguous. No other country name contains the word America, and no one else commonly refers to themselves as American in English.
I don't go around telling Colombians they can't call themselves Colombians because Columbia is another name for the Americas. It's also not a problem here because we don't recognize one American continent.
Who refers to themselves by continent anyway? In what sense is it useful to identify to yourself as someone who lives somewhere in the entire western hemisphere? There is also no common language, race, ethnicity, or culture that unifies North and South America.
satori99 Waves hello from Australia :)
By the way, do people from New Guinea get on your case about calling yourselves Australian (since it is part of the continent), like South Americans do to us?