No other country has the name 'America' in it, only the United States of America. In fact, America is the only proper noun in the USA's name; both 'united' and 'states' are rather ambiguous. Furthermore, we're talking about nationality, which has a direct cognate with 'nation'. North America and South America are not nations; they're continents. Therefore North American or South American are not really nationalities.
What really bothers me is when people refer to Native Americans as Indians. Because the United States only called itself America once it was colonized, I'm not bothered by European immigrants calling themselves Americans, but calling Native Americans Indians is ridiculous as there's no correlation between Native Americans and the country India.