There is no party even capable of doing it.

The Gulf coastline is almost 1000 miles long, there would have to be a gigantic occupation of an area the size of a small country, at the same time as there would be 'all out war' with Iran, which would be backed by China and to a lesser extent Russia, and whereupon an invasion would provide them with millions of determined fighters.

We're talking 'Gulf War' scale of operation against a much bigger, more capable country, and of forces willing to fight.

And the US doesn't even have anywhere to do it from.

Assuming a Gulf country would host an invasion force - extremely unlikely - there's no magical way for US to cross the Gulf with large numbers of forces, as we can't get capitol ships in there in the first place.

There's no amphibious capability at the scale necessary on the Arabian Sea.

Literally just the logistics of large scale landings is almost impossible.

That leaves the Kuwait / Iran border, and maybe something a bit wider.

And then fight through the mountains across the Gulf?

The thought is absurd, it's a 'major campaign theatre' - of which US forces were theoretically capable of fighting in two at once, but that's not pragmatic. That's 'wartime economy' kind of thing.

It's possible but unlikely that 10K marines and paratroopers are going to be able to do much, because it's very risky and likely won't accomplish much.