Davies would probably have an objection to that term as well.
After all, the people who we now call Native Americans, besides not being unified in culture or origin or time of migration very much at all such that a singular term like that is not accurate anyway, started migrating to the western hemisphere about 7 - 15000 years before Amerigo Vespucci was born.
Remember, these arguments are from a historian's / archaeologist's viewpoint. I am sure they dramatically prefer the academic designations for various cultures.
They certainly didn't call themselves 'native americans'. That's a label we slap on them post-facto.