According to "Court Martial", his specialty is space psychology - a very useful skill to have on a five-year mission, and probably it makes him a good choice for Starfleet to replace the Piper/Dehner combo. Perhaps such concerns outweighed the need to care for Spock's well-being?
They say he's an expert, not that it's his specialty. McCoy actually downplays that by saying he knows something about it. If it was his specialty I don't think he'd do that. Wouldn't Dr. Noel be aboard at this time or did she leave after Dagger of the Mind