I can't help but feel you aren't "getting it".
He posted this to a group that had over 270K members. Both the individual and his employer were easily identifiable. It is entirely possible/likely that nobody "reported" him and that people in his organization saw it for themselves.
He was at work, on property, discussing patients in a public forum in a way that he intended to be humorous. Problem is, most people would not want their health situation teased about, laughed at, made fun of or used for satire by the individual trusted to care for them. THAT action gave the public legitimate reason to believe that institution is okay with that behavior. Therefore the institution really had no choice but to fire him, to send a message to the public and to their employees that people aren't going to be making fun of/teasing/laughing at them if they come there for care, even if they aren't in their right mind at the time they are being cared for.
That's the material point. It was a serious error in judgment. The firing was heavy handed but far from outside the realm of a reasonable reaction in today's healthcare and social environment. This isn't a freedom of speech issue. Every one of us is free to say anything we want on any platform we want and we can't be put in jail for doing so. We can, however, suffer the consequences of our actions in all kinds of other ways. His firing was a natural consequence of laughing at/about a patient entrusted to his care in a public forum.