I think the question really is, "Does it require an act of Congress to change a geographical placename in the United States?"
I mean, Mount McKinley is in a national park. And national parks are run by the Department of the Interior. The Department of the Interior falls under the Executive Branch.
So wouldn't any reference to the mountain that would be on a map that comes out of any Executive Branch agency... fall under the President's purview?
From what I could tell, it doesn't sound like anyone is forcing the National Geographic Society to change the name.
I'm not sure you can believe this and agree with NavyCanDo up thread.
But, avoiding any comment on Trump, I will say this.....
It was an act of Congress that changed it to McKinley, and I would guess that Punk Dictator Obama hasn't got a clue as to what his role ought to be in this. He's just trying to make noise, get another voting block, and distract from his economy-killing capitalism-hating climate change moves in Alaska.
And he thinks he can do whatever he jolly well pleases.
(And apparently is right about that.....
)