They are the dominant race. They just don't assert the dominance.
Here is an analogy for you. Humans have had the ability to go down into the depths of the ocean for many years. While we're down there, do we go on a killing spree to take over? No, we're just there to explore. We are the dominant species. We could kill everything down there, but what good would it do? We'd exterminate several useful species.
How does this relate to stone men?
Well, it is quite simple. They need not breathe, and they are essentially 'immortal' and are very strong. Why would they have to live on the surface of a continent? Perhaps they actually live in the oceans and deep underground. After all, lava might not be able to kill them either so maybe they choose to live by lava springs? Regardless of that, my point here is that the surface of the land is to stone beings what the surface of the ocean is to sharks and fish. Let's assume ocean fish were intelligent. Aside from when humans go on a killing spree, why would fish attack ships? Fish don't attack them now, so there is obviously no territorial or survival aspect to fighting over the surface of a 3D volume.
Now of course there arises the issue of why they don't become aggressive conquerors. Well there are a few reasons this might occur.
They don't know of continents.
They do know of other continents and live in them but seriously don't care about surface dwellers.
Now, we can say that stone men can move to other continents. All they have to do is tunnel beneath the ocean. It isn't that hard. They just have to know of their existence. I've already explained why they wouldn't attack countries they discover. After all, fish don't attack boats for no reason, nor do they declare war on pelicans. If there's a conflict the fish can easily just swim down deeper. Same with stone men.
With this in mind, let us assume the stone men were aware of the cataclysm but unaffected. Well, this is completely comparable to the issue of refugees and human countries. Now if we presume that this is the stone mens' solely owned continent they have a few options on how to react to what are likely defenseless refugees. There might be a few who could kill a stone man but we'll assume that with decent strategy they could wipe out the humans.
Reject the humans at the border. Countries don't declare war on immigrant refugees.
Make them citizens of the country subject to the same (applicable) laws as stone men. If they pay taxes to the stone government and aren't thieves why would the stone men care? Literally, the only argument against this is racism and it is unlikely that any race would be that unified in racism. Even then, the issues here are likely to be more in the realm of serial killer stone men, discrimination, and general avoidance of each other. After all, the racism could go both ways with groups in both races falling either way.
Complete ignorance. Stone men only use the surface for things like venting pressure underground and cooling/warming regions or collecting water for various crafts. In other words, the surface is only useful for scientific resources. They don't eat and they don't need heating or anything. Wood is probably useless as it is so flimsy. The irony here is that trade with the humans would show it is more useful to humans than stone men. In the end, there is no reason for the stone men to even interact with the humans. Note however that this isn't due to reasons of superiority or wisdom. The reason they ignore each other is because they live in fundamentally different environments.
Imagine how many houses and cities you can build within a continent with tunnels and caverns. Now imagine how many cities you can built on the surface of a continent. I would imagine that the surface homes are probably for wealthy stone men. After all, to them it is a limited space with amazing access to wildlife, sunlight, and coastal views. The humans are literally living in "luxury housing". So if this is the first meeting of stone men and humans, the interaction will probably go as follows (printed on stone tablets of course):
"BREAKING NEWS: Luxury housing being shut down due to flesh men influx. In other news, president Stoneface is passing a new gold regulation bill."
"Wow Charles, did you read the stone news? Those flesh men are really sticking it to those rich guys. I was sick of them acting all high and mighty up there."
In other words, the humans are literally taking property from rich stone men and using it for refugees. What society wouldn't find that ironic and laugh it off? All you'd have to do is show that the rich stone men get a lot of flak for for political actions. After all, who is going to rush to the aid of dictators or jerks complaining about what is essentially a "pest problem"?
For that matter in an ironic twist the surface then becomes the "poor housing" and all the stone men are even happier because rich folk are so bent out of shape about humans that they practically gave it away to people who could never afford.
I know this answer was quite chaotic. It is kind of a train-of-thought and I think I suggested a lot of things worth thinking about here. Ultimately, this is an issue of understanding the politics, racial queues, exploration level, economy, and societal attitudes of the stone men.
And as a final nail-in-the-coffin for the stone men let me just point out the biggest thing that allowed Europe (essentially an island in comparison) to conquer North America:
They're from a foreign land and stone life exists. Why can there not be stone diseases? Humans developed an immunity or maybe they are carriers (stone diseases can only steal nutrients and people just take it as food need disorders and deficiencies). Whatever the case, diseases mutate and grow rapidly. It is their nature and why they are deadly. A foreign land separated for possibly millennia filled with beings immune to stone diseases? It wouldn't surprise me if they bring with them the stone equivalent of the bubonic plague. Hard to fight off a human scourge when you're entire civilization is collapsing from plague after plague. Just saying.
If we presume humans are incompatible with such diseases you can also say that stone life in general exists and was domesticated by humans, whereas stone men didn't do to philosophical differences such as whatever prompted Native Americans to not domesticate wildlife (such as the non-existence of horses and large cattle on the continent). Those types of human-animal interactions enables a higher frequency of mutations in diseases. Hence, the result is "hyper-diseases" which essentially wipe out races that didn't have such interactions. It is literally the most important factor the Europeans had. Otherwise, the Native Americans would've been able to wage a much stronger effort at evicting the Europeans. Who knows? Maybe they would've convinced the various companies building colonies to give up and just trade with them?
I can definitely say the stone men wouldn't domesticate human animals. Simply put, they would be worthless to stone men. They cannot eat the meat and they cannot use them for materials. To them, bone as a tool is like using dried leaves. So they wouldn't have a corresponding defense towards humans.
This answer presumes no assumptions about the procreation of the stone men or lifespan. Instead, this focuses upon stone men that live and die within the same general rate as humans. If they live many years, we can expect wisdom to have a larger impact in decisions (which means a likelihood of more sympathy for humans and more support to just "leave them alone in peace").