1st amendment covers State actors. He didn't commit treason or divulge classified information. If he was disciplined, it could be a 1st amendment case .. but if he's represented by the JAG and not by independent council, it wouldn't be good.
If he is enlisted, he has signed a contract that surrendered his Constitutional Rights for Terms and Conditions found under the Uniform Code of Military Justice (UCMJ). This is essentially Admiralty Law. His rights are the law of the Sea until is discharge.
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