For the most part, no. That complete extraterritorial status belongs mainly to embassies and consulates and a handful of military bases, and even then the idea of them being "sovereign territory" is really a fiction. They are effectively sovereign territory since the host country can't step foot inside without permission, but they aren't really US soil. If you happened to be born in an embassy, I believe you'd still get a consular report of birth abroad instead of a regular US birth certificate.
The rest of our bases operate under various SOFAs (Status Of Forces Agreements) that give different partial aspects of extraterritoriality, but not complete. It would have been a different story if he had been born in a ship's hospital, as I believe all naval military vessels are considered to be the sovereign territory of their country.