I find myself wondering how much management experience you have. There is no implied judgement or criticism, but you seem to be reacting to events in a way that suggests relatively short tenure in management.
When we're triggered by events, e.g., a direct report escalating one of our decisions, we must of course deal with our reactions through reflection, talking to a coach or mentor, etc. Then it's time to act constructively.
One thing that helps is to assume -- until and unless proven otherwise -- that the employee's behavior was valid and reasonable from his point of view. While it may look to you (and me) that he seems naive, short-sighted and impulsive, you could shrug that off as a rookie mistake. If you can get your head around the notion that he was acting as he thought best, and even though he seems to have thrown you under the bus, you get to frame the events as an opportunity to engage with him. Find out his story of events, his actions, the repercussions. Then, if it seems appropriate, share with him a bit of your experience. "I have to admit that I was a bit upset when you went to my boss, and then threw me under the bus. But I'm over that now and trying to find ways we can both learn from this..."
Finally, I think it's potentially powerful to interpret his aggressive moves as healthy ambition in need of tempering in the realities of political forces. Ask him if he's open to your coaching him and helping him succeed and move ahead. Maybe you can help him understand and develop the skills and knowledge he needs to be able to handle YOUR job. Your willingness to help him in this way could be motivating for him, and engender trust. Meanwhile, it's time for you to set your sights on your next position. Maybe your boss would be open to helping you.