"Indigenous" is the word you're looking for, not "native".

I've seen no evidence that the term "Native American" is in any way a term Indians use to describe themselves. It's most likely a neologism of white origin used to assuage white guilt. It's considered respectful to call any group of people by the term they prefer to call themselves, not to invent your own term out of patronizing guilt for 400 year old navigational errors.

(Incidentally, "native American" is ambiguous in that it could also be used to describe people of any race who were born in America. In fact, the term "native American" was used in this manner throughout the 19th century by anti-immigration activists, also known as "nativists". That's why "indigenous" is the most general and ambiguous term.)