As far as I know, calling ourselves American goes all the way back to the beginning of the country. I can just imagine the founders sitting around with some beers trying to figure this one out...
"Guys, we need a name for us since we aren't Europeans anymore- what do we do?"
"Well, we could be United Statians."
"No, that's stupid."
"How about we just all go by our state affiliations?"
"We're United, remember?"
"Dammit, let's just call ourselves Americans- it's easy and there's no one else here using it."
"Sounds great! Say, gimme another round."
I think they called themselves: "We the people of the United States"