Is it true that most Native Americans (USA) are still quite pure?

Those classified as “Native American” are not pure in the context you’re using.  They are a mix of the Aboriginal Indigenous Americans with Mongolians, thus creating mulatto who later procreated with “white” colonizers and slaves and continue to this day. They look ambiguous like most mixed race children. These people are wards of the state. 

Sadly, there are relatively few Natives who are full-blooded in the U.S. compared to those who are mixed-race, unlike most other people groups. If things continue on the track they are on, with Natives only caring about their cultures being passed on and not their genetics, and the seemed majority mating out, first a part of human genetic diversity, and the beautiful Red race, will be wiped out, and eventually their cultures could be wiped out as well.

Your question intrigued me, so I googled. And I've been reading articles on the subject for half an hour and my conclusion is that no one really knows. There's been mixing in every part of the country, more in some parts than others, but without doing DNA tests on everyone who describes themselves as Native American, there is no way to tell how much. I'd be pretty sure, however, that there has been less mixing in the western parts of the country for various reasons, one of them being the mere passage of time. The eastern parts have had a couple hundred more years for the mixing to happen. I invite you to do some googling too.

In Alaska and the southwest, native people are less mixed

"pure" is a euphemism for "inbred"

For tribes that were almost extinct, the choice was between "mixing" and "inbreeding"