Indeed. 12 nautical miles from the low water baselines are sovereign territory per UNCLOS. Same as if it was land. The next 12 miles beyond that is called the contiguous zone. They are also sovereign, except other nations have the right to overflight. Then there is the 200 mile EEZ, where Iran owns the resource rights, but other nations can sail through. It is all divided up by adjoining nations, with the median rule applying where overlaps occur.
And here (attached) is the US Dpt of State document from 1994 recognizing Irans claim under UNCLOS (Although neither Iran nor the US ratified UNCLOS, the claims are recognized). And yes, the US disputed the low water baselines claimed by Iran (they are using straight line segments, same as China, that goes against UNCLOS guidance), however the sovereignty is not disputed, it's the extent that is.
It's an odd technical feature on UNCLOS, that the UN has no power to decide on the claims made by adjoining States. Called delimitation. It is up the adjoining states to mutually sort it out. The US can lend support to either state, but they have no official part to play unless it is their coastline and water.
It's a concept the UN calls " Land has dominion over water". That is the basis for UNCLOS. That to control any part of the sea, the state must have land there. And in this case, Iran does have land there. The USA does not,
LIS No. 114 - Iran Maritime Claims (state.gov)
Page 15 of that report is the important bit. It covers Iran's claims to article 8 and 9 of UNCLOS. Suspension of innocent passage and Exceptions to innocent passage. Except when I check UNCLOS, the US got that wrong. It is not UNCLOS article 8 and 9, it is articles 17, 18 and 19.
UNCLOS allows for exceptions to the rules, but only if lodged and published at the time of signing. Iran has signed, but not ratified.
Sorry for the waffle, it just happens to be a subject I have had an interest in for a fair number of years, because I follow the South China Sea dispute. It's a really technical thing that I struggle with at times. Case in point, see bolded part above where the US state dpt got it wrong.
I am not any sort of Lawyer, just a layperson with a long standing interest. And to get a good grasp on the SoH one really needs all the docs and detailed maps.
And no, the above is not AI.