From what I understand Africans did not call themselves "Africans" either, and I'd bet that it's the same with Asians or Native Americans, who probably just call themselves by the name of their tribe. It's really only Europeans I think who call themselves that, it makes sense since it's the smallest continent AND the one with the most external contact (Africa to the South, Asia to the East, and Americas once they were "discovered"). So it makes sense some kind of "common identity" would develop early there, even more so that most of the continent was dominated by various empires at the same time (Romans, Christianity, later on Napoleon etc).
Now "Native Americans" have to call themselves that way because citizens of the United States has usurpated the term "Americans" from the whole damn continent, meaning other have to specify what kind of American they are ("Latin" American, "Native" American, "African" American). In french there are two terms : the widely used "américain", and the hardly-ever used "étatsunien", which is much more specific to the US ("Etats-Unis"). I think there isn't even another term than "American" in english.