Pretty much any mapping software will follow suit as It's an official name change, so until it's reverted (if ever, because I could see a future Democratic President just leaving it as is because they don't want to be called anti-American on Fox), you'll just have to deal with it, unfortunately.
OSM treats official names as tertiary to signage and local usage. It's also not chiefly American so doesn't have much reason to favour their usage over other countries'.
Mind that you can also have many names for one thing in OSM so it will probably be noted in there somewhere.
I had to have a look at the current situation, Gulf of Mexico currently has no fewer than 39 names listed
There is also an ongoing discussion about the executive order. From what I saw it seems to be leaning in favour of adding an official_name:en-US tag if the order gets recognised by a relevant body e.g the US Board of Geographic names. Notably this is not the primary name tag.
I guess the Gulf of Mexico is different from Denali in that it's outside of the US, so I'm not sure how that's decided. I guess American map makers (or anyone localising for the US) will use Gulf of America, while everyone else will continue using Gulf of Mexico.
FWIW, it's still Gulf of Mexico in my Google Maps, and so is Denali (using it from Australia).
It's not an official name change... They have a governing body for name's and unless they change it, which they won't only the us will recognise the change... Whole things a pissing contest like trump himself... He's a bully and not a very good one