"I hereby find that the immigrant and nonimmigrant entry into the United States of aliens from countries designated pursuant to [law], would be detrimental to the interests of the United States, and I hereby suspend entry into the United States, as immigrants and nonimmigrants, of such persons for 30 days from the date of this order."
I don't know just how entering with a green card qualifies, but "immigrant and nonimmigrant" would seem to cover everything.
As far as actually having the power to do so, everything I've read indicates that the President has extremely broad powers to determine which noncitizens are allowed enter the country.
I would also question if the President can ban legal permanent residences of the country from re-entering simply based on where they were borned.