In concert with the Congress, he certainly could, if he was disposed to do so.
After all, if the national legislature and executive had nothing to say about these matters, they couldn't have forced Utah and several other Western states to forever forsake plural marriage as one of the main criteria for entering the Union in the first place.
even the federal marriage amendment would go only so far as stopping the states from using the word or the institution of “marriage” as being anything other then one man/one woman.
but so long as the States don’t use the word “marriage” to define the terms of civil unions, they can still have their own laws on that. it could not prevent the States from passing a law that allowed Harry and Mike to have health care proxy rights, or rights in wills and estates, etc - so long as its not “marriage”.
The bottom line on that is - so long as the makeup of the electorate in a particular state is zoned out enough to accept this, there isn’t much a President can do to stop civil union style rights in the States.