If Trump really does go ahead with an official effort on this, the aftereffects will be entirely political and will have nothing to do with mapmakers’ opinions or objective cartographic conventions. Compare the longstanding effort to rename the Persian Gulf as the Arabian Gulf, which I had never heard of until my Iranian-born wife told me about it.
The short version is, back in the late 50s into the early 60s, as Middle Eastern countries began leveraging their petroleum wealth and emerging on the world stage, pan-Arab activists were advocating for regional alliances that would allow them to maximize their power and influence. Iran, for historical reasons, was the odd country out, so the idea arose to rename the Persian Gulf, long known as such, as the Arabian Gulf, as one of several tactics for marginalizing Iran in the region and inflating global perception of Arabian identity.
A few countries signed onto the proposal, but then it never accumulated wider acceptance beyond that. Nevertheless, the countries that did adopt the new name took it very seriously, and have maintained the position to the present day. It has nothing to do with maps, and everything to do with political alliances and diplomatic deference. If you want to do business with or in one of these countries, they expect you to respect the Arabian Gulf convention in dealings with them. If you want to signal you’re a more serious ally, you adopt the new name on a wider scale.
For example: Officially, the US position is to retain use of the “traditional” name in state publications (or one of its variants, like “Gulf of Iran” or “Gulf of Fars”). However, whenever American military operations are conducted in one of the countries advocating for the new name, US personnel follow standing orders to say “Gulf of Arabia” or “Arabian Gulf” when interacting with their hosts. Military publications still refer to the Persian Gulf, but in face-to-face meetings, the host country’s preference for the new name is respected.
Wiki link with more information.
Presumably, if Trump does actually order his people to try to make the “Gulf of America” thing stick, a similar divide will emerge, driven entirely by political calculation.
Edit to add a postscript: Yes, I do amuse myself by teasing my wife with “accidental” references to the Arabian Gulf. She doesn’t think it’s especially funny, of course. 