I don’t know what Romney would do; with Obama, it is a good chance he will not act on the appeal.
The military should quietly not comply and let they gays sue them in court if Obama does not act. By that time it will be a new Congress and likely Obama would be out of office before the dust settles.
I bet 0bama won't do a thing to resist this, but will be glad to point to someone else to take any heat for a decision he'd love to make himself (but if he did people would actually criticize him in public).