From Kevin, There are hundreds of million people in a whole range of countries who are also perfectly entitled to call themselves Americans.

There are also million, even billions of people who are entitled to call themselves Europeans, Asians or Eurasians (for example.) If they choose to do so, they are welcomed to do so. Most people, when referring to their nationality, use a name that reflects the name of their country. As some one above noted, the USA happens to be the only country on either the North or South American continents that incorporates "America" in the name of the country.

If any country in Europe, and only one country on the continent of Europe, used Europe in it's name, I think it likely the term "European" would commonly be understood to refer to people from that country rather than referring to all who dwell on that continent.