I tend to call the USA the United States, but either you can accept the common demonym and adjective of "American" which is an entirely separate thing than the name of the country or you can not. Same for Mexico.

You can choose how you want to address people of the USA, but it's up to them to choose how they react to it. "Yankee" is another alternative that's a lot easier to say, even if they don't tend to like it.