He does not have to.
Executive privilege.
The concept of Executive Privilege has nothing to do with enforcing the decisions of the courts. It's a completely unrelated legal concept.
If you think the President should refuse to enforce Supreme Court decisions with which he disagrees, fine. But just remember that if/when you get a Democrat in the White House. Because you'll have opened a door that can't be shut again.
I'm a big fan of Dubya's. But if this made it up to the Supreme Court, and the President publicly stated that he wasn't going to enforce the decision because he didn't like it, he should be impeached.