Wokipedia says that the Gulf of Mexico/America was referred to at some point by at least some people as the Golfo de Florida. Here’s the cited source with, in turn, some of its citations:
Here’s the section that seems to be the basis for Wikipedia’s “other explorers”:
This seems like a good bachelor’s thesis topic a history major! Separately, if the Gulf of Mexico v. Gulf of America dispute can’t be settled amicably, my vote is for Golfo de Florida!