Let’s clarify the question: Do you mean OWN or RULE?
Nearly all territory within the borders of the United States is U.S. territory, for political and jurisdictional purposes, but, except for the huge national parks and such, very little of this acreage is property of the federal government. My house, if I own it, is my house, and the police cannot enter it without my permission or a warrant. On the other hand, I am not a sovereign (though some Americans nowadays, in the militia and “common-law courts” movements, seem to have delusions to the contrary). My house is U.S. territory and if the police HAVE a warrant there is no way I can keep them out without committing a crime in the process.
The grounds of a foreign embassy in Washington or consulate in New York would in most cases be the property of the government of the country whose embassy it is; perhaps it would be property of some private individual or corporation who leases it to that government; perhaps the property of the U.S. government, leased to the foreign government. In any case, the embassy would NOT be U.S. TERRITORY even though enclosed within U.S. borders – the embassy would be an extraterritorial enclave, and no American soldiers or law-enforcement officers would be allowed to enter without permission; such impermissive entry would, presumably, violate the treaty establishing diplomatic relations between the U.S. and the foreign state, and might even constitute an act of war by internationally recognized standards. When Noriega took refuge in the Vatican Embassy in Panama City, U.S. troops couldn’t go in and get him without the permission of the pope or his ambassador/nuncio. (I forget how they resolved that – I think after a few days of amplified rock music coming in through the windows from sound trucks in the street, Noriega ultimately came out, of his own more-or-less free will.)
If the above thread is correct, that Rockefeller gave the U.N. the lot its New York HQ is built on, then the U.N., as a corporate entity, owns the grounds and building, free and clear. But that still leaves open the question: Does the U.N. headquarters, as such, have extraterritorial status, such that NYPD officers or U.S. soldiers could not enter (in their official or military capacity) without permission of the Secretary General’s office? The obvious answer would be yes – since the U.N. is an entity with which we have diplomatic relations and to which we send “ambassadors.” On the other hand, the U.N. is NOT a government or a sovereign state. In that respect, its political status is lower than the Vatican’s. It might not have the right, under international law, to rule or control a single square foot of territory, in a political sense. Does anyone have a theory on this point?