Ignoring the fact that there’s no such thing as a “native white American,”...Well, there is, actually.
The word "native" isn't inherently problematic, but by the time I was in high school I'd noticed that it could be used in ways that set off alarms. One is exemplified by "the natives are restless," which was already something of a joke in my lifetime, and I'm pleased to see still is. At its root, though, it's a paradigm example of Othering, of dividing humanity into Us (the civilized white people) and Them (the dusky locals, the White Man's Burden). To call someone else "natives," then, was racist.
On the other hand, I also encountered the term "nativist" in American History class, where it denoted anti-immigrant groups and sentiments -- ironically enough, these groups were usually made up of people whose recent ancestors were immigrants themselves. So one could refer to oneself as "native" and mean it positively, but nativism was hardly an ideal I was going to aspire to.
So I've always been wary of the term "Native American" for people descended from the pre-Columbian societies of the Western Hemisphere. I hoped vaguely that it was meant at least somewhat ironically, with some awareness that the word "native" is double-edged. The more I've heard the term used, however, I've had to let that hope go.
"Native" comes from the Latin word meaning "to be born," and means primarily that one was born in a given locale. The blogger I quoted above, like many who use the word nowadays, means something else: apparently, not only that you were born somewhere, but all your ancestors were too. The idea would be that the species, or the "race," is viewed as a single entity, which originated in a specific place. If that's the case, however, then the First Nations aren't natives of the Americas: their distant ancestors came from northeastern Asia, most likely, but they weren't native to that part of the world either. As far as we know, the human species is ultimately "native" in this sense to Africa.
So the blogger's claim that there's no such thing as a "native white American" is not only false on its face -- I'm white, I was born in Indiana, hence I'm a native white American -- but it's racist. When a white American asks a non-white American, "Where are you from?" and refuses to accept "I'm from here" as an answer, perhaps probing for the country her ancestors came from, most people would recognize the intent as racist: Even if you were born here, even if your parents and grandparents were born here, you don't belong here (but I do). And it is racist. So is the blogger's claim.
I suppose one could argue that "native" ought to refer to the origins of one's ancestors, that there is some point at which one's lineage ceases to be "immigrant" and becomes "native." I'd like to see the argument, though, and I'd like to see the justification for drawing the line wherever such an argument chose to draw it -- how many generations, how many years? It will, I suspect, always be arbitrary. (I wonder if some Ojibwe culture cops objected to teaching Darwinian evolution at a tribal college, not just because "Nothing in our oral traditions says that we came down from trees" [Scott Richard Lyons, X-Marks (Minnesota, 2010), 96] but because contemporary archaeology puts human origins in Africa instead of the Americas.)
"Aboriginal," as Merriam-Webster defines it, seems to work better than "native" for this purpose: "being the first or earliest known of its kind present in a region." It doesn't have to imply that human beings originated in a locale, only that the first human beings present in the region were of this group. This still has problems, since the migrations from what is now Siberia probably occurred over time in successive waves, and I daresay they weren't always free of conflict. Besides, "aboriginal" has also acquired racist connotations ("primitive"?). I think "native" has strong associations that are inseparable from its racist overtones, because despite our tendency to move around, human beings also like to feel tied to place. "Native" can be a neutral term, especially when it's used as an adjective rather than a noun, but one must be on guard against its racist connotations. The trouble is that for many people who use it (including the blogger I quoted), those racist connotations are part of its appeal. I belong here, you don't. I hold that wherever you feel you belong, you belong; but belonging doesn't entitle you to exclude the belonging of others.
What does this imply for the ongoing struggle of First Nations to preserve their cultures and lay claim to their lands? I don't think any term is essential to these struggles, and I increasingly believe that "Native" was adopted partly for rather than despite its racist associations. My objection to "Native" doesn't mean that I minimize the crimes committed during the European invasion of the Americas, or want to deny the validity of the claims American Indian activists are making; none of these matters hangs on one word. But when your chosen terminology leads you to lie, as with "there's no such thing as a 'native White American,'" it's time to reappraise your terms.