"Gulf of North America" would make more sense, because most people regard (just) "America" as synonymous with "USA". Though that name would perhaps be more appropriate for the Hudson Bay, which is farther north.
That's not true. I live in Europe(Germany) and I as well as everyone I know personally says "America" when we mean "the USA" and mean "the USA" when we say "America".
I visited Germany a few years ago and I can confirm that they called me “The American” and referred to the US as America. My colleagues explanation was “There’s Romanians, from Romania. There’s Hungarians from Hungary. There’s Americans from America.” I laughed a bit when he said it.
"Gulf of North America" would make more sense, because most people regard (just) "America" as synonymous with "USA". Though that name would perhaps be more appropriate for the Hudson Bay, which is farther north.