The name of the water body as recognized by the United States is set by the US Board of Geographic names (https://www.usgs.gov/us-board-on-geographic-names/domestic-names). That entity meets occasionally and releases data that is displayed by the National Map. Here’s a link to their entry for Gulf of Mexico: Geographic Names Information System.
Even if the name changes there, I would argue that OSM would specify it as an alt_name, not name:en or even name:en_us, since the majority of use in English or American English is still “Gulf of Mexico”.