No. The whole secret stargate agency, the one in all those bunkers, was a creation for the series. The original movie didn't involve a giant secret operation. It was a basic take on the whole 'ancient aliens' concept. They expanded everything to make room for the series long after the name. I think gate was more to do with star trek and wars, but by that same logic the government program could be named after trek and wars. The people picking names for secret government programs like to play jokes. They are all scifi geeks too. See Skynet and Reagans Starwars.

I'd be 100% behind calling the new 'space force' the Starfleet. There is no harm in making a name fun.