At PCA we promote the ROOTS of Honoring the Game (respect for Rules, Opponents, Officials, Teammates and Self).  Your player's public criticism, behind your back, violates respect for Teammates -- which includes the coaches -- in a very dramatic way.

Rather than responding in kind, model how you want him to behave and demonstrate your maturity by having a private conversation with your player.  Let him know why this behavior is unacceptable and how it undermines the potential of the players and coaches to become a strong team, which requires the development of trust.  You might let him know that if he is willing to throw you under the bus publicly, his teammates may worry he could do the same to them.

Often coaches are reluctant or even afraid to hear suggestions from their players.  If you feel comfortable doing so, you might encourage him to share his concerns with you rather than complain publicly through Twitter. 

"Look, it's not okay for you to criticize any member of the team publicly, including me.  I need you to stop doing that.  But I do want to hear from you if you think we can do things better.  Part of being a good teammate is to help each other get better.  So if you have a concern about the way I am coaching the team, I'd like you to tell me privately rather than Tweeting about it.  Can I get your agreement on that?"

Strong leaders welcome suggestions from their followers even when they are hard to hear.  As Jim Collins, acclaimed author of How the Mighty Fall and Good to Great, noted in his keynote at PCA's National Youth Sports Awards event last month, great organizations have the courage to look at problems rather than paper over them.

Opening the lines of communication this way doesn't mean that you have to agree with everything you hear.  But if you can find it in yourself to encourage your players to share their concerns with you, you will have the chance to communicate with them in a way that could lead to a wonderful feeling of team and improved performance. And being listened to by you can fill a player's Emotional Tank even if you decide not to take his suggestions.

For the future, I encourage you to introduce your players to the idea of  Triple-Impact Competitors who are committed to improving themselves, their teammates and the game as a whole.  Then when something like this happens, you can refer to how this kind of behavior violates the commitments of a Triple-Impact Competitor.