The gynaecologist was not unprofessional with the prospective patient, he stated that he did not have the expertise to treat the patient, and offered to refer the patient to the appropriate services. He used female pronouns.
So where's the unprofessionalism in that?
The patient became loud and abusive.
The patient's partner later posted a comment that was critical of the doctor's behaviour to the patient in person, despite there being no evidence that the doctor was anything but professional towards the patient.
If the patient's name was revealed on a publicly available site, it must have been revealed by the patient's partner, because the doctor's post does not name the patient, and his comments are not 'highly personal'. He has apologised for the emotional tone of his reply - which I don't think he needed to do - but his apologies have been ignored.
It really does look like a set-up, as BunfightBetty says; the patient must have known that a gynaecology was not the appropriate specialism for them, unless they are very very ignorant of human physiology, including their own.
I would also worry about how he responded to any woman making a complaint about the treatment she had received.
Dr. Acharian is a gynaecologist. Obviously, a woman patient is never going to place him in the same position, a woman couldn't make such a nonsensical complaint because it's logical for a woman to see a gynaecologist.
It's pointless to speculate on how he would respond to a complaint from a woman because it would be entirely different.