she was harassed and threatened (including death threats) by RW readers of his blog.
Considering how often that sort of thing happens when a RWer with a public platform sets up a designated liberal for public harassment, giving harassers enough info to enable them to go after her was ill-advised at best. At worst, he might have wanted her to at least get hassled and embarrassed, even if he didn't realize that death threats would ensue (though how anyone could not realize that these days is beyond me!).
I imagine that is part of why he is suffering such consequences. Also, as a tenured professor at the university, he has certain responsibilities toward those who are students, including graduate students who teach low-level classes. TAs are considered to be "in training" as teachers. If he thinks she handled the classroom situation badly, then he should have spoken to her supervisor in her department, not flayed her publicly.
In fact, even though he is not in the same department she is, I could see this as being considered a violation of the Buckley Amendment, which protects students' privacy. Interpretation of that amendment has always been strict, so strict that we cannot even to send our college students information about their grades by email, because email messages are not secure forms of communication. Nor are we allowed to just throw out old tests or essays without first shredding them, since someone might find a student's essay or exam in a landfill somewhere with the name still legible.
Since she is a student, and since a TA is a teacher "in training," a senior professor's publicly complaining about her by name would probably be considered a pretty serious violation of that amendment.