Interesting; I had assumed the executive order was intended to make the name apply to the entire gulf, but the Mexican President’s phrasing of “stick to what the United States government approved” seems to contradict that, so I looked it up, and indeed it does acknowledge international boundaries within the gulf (emphasis mine):

…the Secretary of the Interior shall… rename as the “Gulf of America” the U.S. Continental Shelf area bounded on the northeast, north, and northwest by the States of Texas, Louisiana, Mississippi, Alabama and Florida and extending to the seaward boundary with Mexico and Cuba in the area formerly named as the Gulf of Mexico.