He could, just as easily as he tried to eliminate gays from the military. It would be just as stupid politically, and if he has the sense God gives a toad, he will stay as far away from the issue as he can arrange.

When I was in the military, I was told that it was OK to disobey an order, but you’d better have a real good reason for doing so.
Congress, as well as the Supreme Court, could overrule Trump’s order.

The only justification for flying a Confederate battle flag at a military base would be in a succession of historic flags that fort or base was under—such as with an historic French, Spanish, Portuguese flag, etc. in historical succession beneath the current American flag. It is doubtful that a U.S. President could break such longstanding tradition, or would even want to.