At about what year in the history of what is now the USA was the number of "native Americans" there exceeded by that of those who were not ?

Fred, any year offered would only be a guess, since there is really no agreement on the number of indigenous people in North America prior to Columbus. I think it was probably sometime near the middle or end of the 19th Century.

This person might know, should you need the information:

Claudette E. Bennett, Chief
Racial Statistics Branch
U.S. Census Bureau
4600 Silver Hill Road
Washington DC 20233

Historical Census Statistics on Population Totals By Race, 1790 to 1990, and By Hispanic Origin, 1970 to 1990, For The United States, Regions, Divisions, and States

Of these, the first I find that accounts for Native Americans is in 1860. However accurate it may or may not be, the Native American population for this Census was but 44,021 out of 31,443,321. The counts include some territories not yet admitted as states.