Do South Americans actually identify themselves with being "South American"? If you ask a Brazilian their nationality do they say South American or American?
How should Mexicans refer to themselves? "Mexico, officially known as the United Mexican States (Spanish: Estados Unidos Mexicanos)" http://en.wikipedia.org/wiki/Mexico So calling us the United States could be equally confusing.
The fact remains that America is the defining part of our name.
Sorry, this started off being a serious question about identity and I got carried away.