Where is the name of that troublesome body of water between Florida and Mexico officially written down? The water between Iran and Dubai is called Arabian Gulf by Arabs and Persian Gulf by Iran and I guess most of the rest of the world. At some point Iran got backing. But one of the most important EU nations is Deutschland and despite it prominence, it is called Tyskland, Niemcy, and Allemagne by its neighbors and if you ask a random American about any of those names, they may not know what country you are talking about because we call it Germany.
I am pretty sure we can just start saying Gulf of America. Whether it sticks depends on whether other people repeat it. I don't think anyone is in charge.