That's a complicated topic, but no. The native governments most certainly assert that they are sovereign nations, and US law does largely agree with that, when the US actually cares to follow its own laws on this topic.
The
recognized tribal nations are considered sovereign - they have the right to pass their own laws separate from state or federal law, to operate their own governments, and not to be superseded in the municipal/state/federal hierarchy that US law has. At the same time, crimes involving non-natives on tribal lands are elevated either to the federal government if one party is a tribal member or to the relevant state government if no parties are tribal members. The native governments have immunity to most lawsuits just like the US federal government does.
This is why you have the classic example of native governments operating casinos - the states surrounding them ban gambling, but their law means nothing on tribal land. The sovereign immunity means even most lawsuits in federal court can't be launched against them. You've also seen this crop up with the covid lockdown, where the Navajo nation (the physically largest native government by far) enacted a total lockdown and road blockade, which they have not lifted since even with the surrounding states threatening to use force against them for it. For once, that's a bluff - the state governments could get themselves in serious fucking trouble violating tribal sovereignty, at least for a recognized tribe in the modern day.
Treaties will also sometimes extend this beyond tribal lands, because the states and federal government back in the day sure liked to make wide-ranging promises. One of the cases where people detected Gorsuch's bias in favor of native rights was one such scenario,
Herrera v. Wyoming, where the Supreme Court affirmed that members of the Crow Tribe had the right to completely ignore Wyoming's hunting laws (out of season, no tags, whatever) under the 1868 Treaty of Fort Laramie so long as they were hunting on unoccupied land, because that was what the treaty promises them. And national forests count as unoccupied.