They (Indians) emigrated here, displacing those before them, just as my forefathers did. Does that make me a native-American?
I know whites who were born and raised in Africa and then became American citizens. Does that make these folks African-Americans?
As for Native Americans, I suppose if you go back far enough, dinosaurs were Native Americans, and before that, protozoans were Native Americans.
What other name would you prefer for "Indians" who were here before Europeans? I don't have a better name.