The FAA could deny approval for launch, but its unclear what the basis would be. What happens on Mars is out of it's jurisdiction, literally.
The UN governs Earth, not Mars. It could expand to cover Mars I suppose. It's the sort of pointless activity you could imagine the European Parliament obsessing over, but there is little they could do about an American launch company.
But the reality is we are centuries away from something approaching a land-rush on Mars, or actual competition between nations for control of anything.
I suspect it will be a "watch and wait and see" situation.