Anyone can claim anything. I can claim the entirety of the world as mine and ‘Boom!’ It is thusly claimed! Welcome to the Free State of ParaSenoia. The question you really have is what does that claim mean? Largely it means that I exert some sort of control over it. Usually we mean political control, but sometimes cultural control or racial hegemony. We also sometimes mean that they have some sort of moral right to the land. Since obviously, the first two only apply in regards to Native Americans to very small regions within North America we can really only focus on the last. The problem with moral claims is that they are largely opinion and they hinge on whatever the person observing the situation feels about it. There might be some sort of objective morality regarding land claims, but most major religions are pretty iffy on it.
Current thinking of ‘claim’ generally comes from Christian thought, notably the ‘Doctrine of Discovery’ based largely on the Romanus Pontifex papal bull of 1454 (and added to later by the Inter Caetera.) The essential idea that was later interpreted into it was that the Church was ultimately the decider of borders and that non-Christian nations did not enjoy the protection of the Church, so whichever Christian nation found it, got to keep it. Obviously, this was problematic, but it largely informed the thinking of the conquest of the Americas (Despite being written 40 years before Christendom even knew about the Americas. It was written to solve issues about North African claims between Spain and Portugal and was largely written in response to wartime conditions between Muslims and Christians and had very little to do with natives, but I digress.) Later, American courts (notably the Supreme Court case of Johnson v M’intosh) enshrined that doctrine into US law.
Of course, Christians being who Christians are- when you put three of them in a room, you get four different opinions, so this is not a universally accepted moral claim for obvious reasons. Some Protestants as an example loved this idea and embraced it pretty whole-heartedly. Some were ambivalent and some thought it was a horrible thing. Nowadays, Christendom as a whole has largely rejected the idea of the ‘Doctrine of Discovery,’ but doesn’t really know how to rectify the situation. The cat’s kind of out of the bag. It’s not like every American of European descent can just pack up and go home. Most of us don’t even know where ‘home’ is. There are also five million Native Americans and 320 million everyone else in the US, so it seems a bit crazy at this point to start evacuating Manhattan. Moral thinking about ‘claim’ has become much more pragmatic among the religious. There’s a fairly large consensus that Natives were screwed. There’s a smaller, but still majority view that they are still being screwed and that the appropriate response is to invest in economic development and help address old claims where practical.
Anyway, I got really off-topic, but it’s only to say that a ‘moral claim’ really isn’t a defined thing. If you as an individual think that Grugg the first guy who made it to America and his descendants are the only people with moral claim to the Americas, then good on you. That’s as legitimate a stance as any. I would say it would be a very tiny minority stance, but if that’s your line, then that’s your line. My personal stance is that it’s complicated and who owned what prior to the Europeans showing up was constantly shifting. Re-litigating the past is neither practical nor particularly helpful. A better stance is what’s happening now and is it just? My answer is that largely it is not just. The reservation system is broken and steps need to be taken to address Indigenous Rights and provide economic and educational opportunities to Native peoples. I’m not particularly interested in who ‘owns’ Mt. Rushmore, I am much more concerned with why Ziebach County is consistently one of the poorest counties in the country with a per capita income that is the same as Angola’s and what we should be doing to address that.