Thanks for the weighed and considered response, and for not closing the thread because things got off track for a bit there.
Now, to put them back on track. Some time ago, I replied with a post that was meant to speculate on what would happen if Europeans did (somehow) arrive in 1492 and encountered people who looked a lot like them. Obviously, not everyone agrees with my speculation on how that would play out, but it does provide an answer to the OP. It takes the premise as a given and runs with it. I'll now take a shot at the other side of the issue: a discussion of plausibility and possibilities. I want to preface this by making clear that I'm not an expert, but that I like to read a lot, and have tried to keep up with the relevant scientific findings. The thing is... Regarding the migration of the first Native Americans (often called "Paleo-Indians"), very little is set in stone. Hypotheses that were considered very well-supported have been put on shaky ground by new discoveries. This has happened more than once, and may well happen again. Previous "consensus opinions" have been demonstrably incorrect, and at present, there isn't even a consensus. There is a majority opinion, but various minorities views exist that have serious academic support. Any of them can be correct, or the truth may combine aspects of these competing models.
I've noticed that some people in this thread have taken certain models/theories/hyptheses, and presented them as being near-certainties. What I'm going to do is to lay out the various models (very generally and roughly) that are (as far as I'm aware) defended by any serious academics at this time. What I hope this will demonstrate is that what some people implicitly assume to be The Facts may not be quite so certain. In fact, there's pretty much guaranteed to be a competing model that also has serious, credible and dedicated supporters in academia. At the end, I'll go into the consequences this all has (or can have) for the subject of this particular discussion. (I stress again that I'm a layman, not an expert. My summation of the various competing theories is bound to be overly simplistic.)
First, we have the long chronology theory. This model is based on the idea that there was an early population of settlers, possibly present before 40,000 years ago. It has been suggested that this wave of migrants may have been related to Australian Aboriginals— a product of the same early wave of human migration. If such a population existed, they were pushed aside or assimilated by a later second wave of immigration. Some genetically distinct communities may have survived in relative isolation for quite some time. (For instance, some think that the original "Botocudo culture" consisted of a population that was ethnically distinct from modern Native Americans, because the ancient skulls are noticeably different.) Since these presumed early settlers were either wiped out or (if assimilated) weren't numerous enough to leave noticeable phenotypical heritage in the later population, we may assume that this would hold true no matter who forms the second wave of immigrants. But the POD "what if no later migration occurred" can give us an ATL where the Americas aren't empty (as some scenarios have it), but populated by distant cousins of the Australian Aboriginals. I'd call that interesting, and therefore worth mentioning.
There is also the short chronology theory, which comes in a number of flavours, which cover basically every other serious model. They are all rooted in the notion that the first migration occurred after the Last Glacial Maximum, which went into decline after c. 19,000 years ago. The old "Clovis consensus" held that the Clovis Culture was the "mother culture" of all Native Americans. It appeared c. 13,000 years ago, which meant that the ancestors of all Native Americans had migrated across the Bering Bridge between 19,000 and 13,000 years ago, via a supposed ice-free corridor. They moved into North America, producing the Clovis Culture, which then produced countless offshoots that went on to populated the Americas. This is no longer the dominant view, although it still has some defenders.
The fact that a substantial number of older sites have been discovered suggests very strongly that Native Americans (of Siberian origins, so not meaning the supposed 'early arrivals') were present at an earlier time than 13,000 years ago. Sites dating back to 16,000 and 15,000 years ago have been found. Those who still defend 'Clovis first' have criticisms of the dating of each of these sites individually— yet the idea that the dating is wrong every time a pre-Clovis site is discovered is quite implausible. The Clovis first model is looking less credible by the day.
Also, the whole existence of any ice-free corridor has been called into question. Even if that corridor existed, though, those older sites are too old to fit with the idea of entry into the Americas via such a corridor. If there was a corridor, the peoples inhabiting those sites must've arrived before it became passable. The idea that they arrived by boat (the "coastal route" model) has now become increasingly embraced. This would also explain why there are sites in South America that are older than one would expect if settlement occurred via migration on foot. But if settlers went down the western sea-board by coast-hugging boats, things make much more sense. This model assumes that the original settlers from North-East Asia arrived quite shortly after 19,000 years ago. (Some academics place their arrival as early as 23,000 years ago, but that's a distinct minority view.) The original migrant population is generally believed to have been small: about 250 people or so.
Does that give us a shiny new consensus, then? No. Because there are still two competing sub-models. The most broadly accepted one holds that the initial 'Bering migration' was followed up by multiple later waves. Three or four waves of migrants from North-East Asia are believed to have moved into the Americas, with the one around 19,000 years ago being the first. (This model makes any scenario where Native Americans of Asian origin are fully absent quite unlikely, as Burton K Wheeler has noted.)
However, there is also a competing model—less widely supported, but certainly not some fringe belief—which holds that there was just one group of immigrants from North-East Asia. Those c. 250 people who arrived c. 19,000 years ago. They form the ancestral population of all subsequent Native Americans. The subsequent 'waves' of migration throughout the Americas isn't disputed, but the adherents of this model believe that those waves originated with the one ancestral population. So instead of Asians making it to America three or four times, they just made it once, and offshoots of the resulting population migrated throughout the Americas in multiple waves. If we assume that to have been the case, it would explain a few things that have puzzled researchers. For instance, one single and small ancestral population would handily explain why Native American HLA profiles are dominated by an unusually small number of types. (Which is one of the reasons why they were so susceptible to epidemics.) It fits. The odds of one population finding its way into America is also a bit more credible then three or four waves of migration finding the way, thousands of years apart.
You can probably tell that I'm a supporter (albeit a cautious and tentative one) of the notion that there was just one ancestral group of immigrants. In any event, the model is credible enough to reasonably serve as the basis for a POD. You can say "that one group didn't make the trip" and you're done. That still doesn't give you any credible way to populate the Americas with anything even vaguely "white", however.
Of course, there is always the Solutrean hypothesis, which claims that Europeans of the Solutrean Culture moved in from across the Atlantic before anyone arrived from North-East Asia. I find the arguments in favour of this model to be pretty lacking, but it's not pseudo-science. Just very unlikely to be true. Both proponents and opponents tend to politicise this theory very heavily. One thing that is of interest is that supporters of the Solutrean hypothesis have pointed out that there is more "Western Eurasian" DNA in Native Americans than one would expect if their ancestors were fully of East-Asian descent. This is all pretty controversial, and studies are conflicting. Also, critics have argued (not unreasonably) that a lot of claims of "white DNA" in Native Americans are based on DNA taken from modern populations, and reflects no more than simple inter-breeding with Europeans as of 1492. Seems a lot more plausible to me.
However... DNA research of ancient human remains found in Siberia (c. 24,000) years old has revealed these to have far more genes linked to Western Eurasian populations than previously thought (instead of being exclusively linked to East Asian populations). Furthermore, it seems that these Siberian remains belonged to people related to the Paleo-Indians. This opens the door to a new hypothesis: namely that the original population that moved in from North-East Asian may have been a mixed group, including people with more Western Eurasian genetic heritage than anyone had previously suspected. This isn't so unlikely: it's becoming ever more clear that ancient nomadic peoples all over Northern Eurasia were highly nomadic, and travelled greater distances than previously suspected. People whose ancestors came from Western Eurasia ending up in East Asia is no longer just a theory. That happened. And it may just be the case that some of them, nearly twenty millennia ago, were among those who made that fateful journey into a new world. If it should turn out that (some) Western Eurasian DNA in Native Americans is older than 1491, this explanation makes infinitely more sense than the Solutrean hypothesis.
I think it's also our best bet to fulfill the OP's idea, insofar as that can be possible. Assume that an ATL group consisting exclusively or near-exclusively of nomads of Western Eurasian descent ends up in the right place at the right time, and makes the journey to the Americas instead of the OTL group. You'd have to furthermore assume that the "one migration" model is correct, but that's not an unreasonable position. Even if the long chronology theory is correct, and there were earlier inhabitants already present beforehand (presumably related to Australian Aboriginals), we may safely assume that they would likely meet the same fate as they supposedly did in OTL.
That's about the closest you can get to the actual gist of what the OP asks. Is this all just as case of "mighty whitey gets there first"? Actually... no. These ATL ancestors of ATL Native Americans might be genetically related to Western Eurasians... but they wouldn't be white at all. As has been observed in this thread, the genetic mutations responsible for that only became prevalent later. These guys wouldn't look like Europeans, and (assuming a butterfly net) come 1492, the Europeans wouldn't recognise them as being related to them in any meaningful way. The revelation would only come with much later genetic research, and would no doubt end up being highly controversial. (The suggestion that some of the original ancestral group of migrants may have had Western Eurasian DNA is already controversial in OTL.)
No matter what, though, speculation on this subject provides major fodder for ATL scenarios. But the idea of Native Americans who are "white", while interesting to discuss as a hypothetical, remains firmly ASB. You'd need to assume the "one migration" model, and you'd probably have to reject the long chronology theory. That would allow you to have "those original c. 250 people never cross into America" as a POD, which gives you an empty America. Then, assuming a butterfly net, you could have the Norse show up as per OTL. They wouldn't encounter "Skraelings", and their little colony might thrive. This would make their descendants the ATL Native Americans. Literally the first people born there. And yes, they'd be white. That's just about the only way to do it, and it takes quite a few assumptions (although not ones that are outside the realm of defensible academic views). I somehow don't think it's quite what the OP had in mind, though...
