Um no.

A state can decide what it names itself or names a part of itself (e.g. Black Lives Matter Plaza). The story of Ukraine illustrates this.

But geographers and cartographers don't decide what to name a place or get orders from states by fiat (unless the mapper is a state agent working for a department) They name things based on what they're called.

The gulf is known to most of the world and the International Hydrographic Organization as Golfo de México or in English, Gulf of Mexico, and calling it the Gulf of America (say by Google Maps) is political allegiance signaling, that they are MAGA or MAGA collaborators.

If you want to be spicy you can call it Chalchiuhtlicueyecatl or the House of Chalchiuhtlicue based on the South American deity of the sea. It has a nice ominous Siege of R'lyeh feel that reflects the tempestuous weather of the ocean expanse.