The concept of America was the product of European settlers. Americans are a distinct and recognizable nationality. Place an American of German ancestry into Germany and he will be instantly perceived to be an American even if he is fluent in German. Irish Americans who go back to their ancestral homeland are seen not as lost kinsmen but as Yanks just like visiting Americans of English or Jewish background. A majority of white Americans are of mixed European ancestry, varying by region. For instance, in the Upper Midwest, there are mixtures of German (including Swiss and Austrian), Dutch, Flemish, and Scandinavian ancestry. The Northeast has mixtures of Italian, Irish, Slavic, and French Canadian ancestry. Appalachia has mixed Scotch-Irish and English ancestry, with some Germans, Scots, French, Irish, Welsh, and Cherokee as well.
The areas of the country largely settled by white Americans of pre-Revolutionary ancestry have a high percentage of persons who self-identify as Americans. This is true in the Deep South, northern New England, and the Ohio Valley. A similar phenomenon may be noted among English-speaking Canadians, who increasingly self-identify as Canadians rather than Scots, English, Irish, etc.
As for the American Indian/Native American, there is no really good term, unless we use the term aborigine, as the Australians do, or Original People, a term used by the Canadian government.