Not necessarily, as he would have been under rather controlled conditions and might well have been denied access to patients who might have compromised Vulcan secrets.
In any case, it's beyond miraculous that McCoy would know the physiologies of two
species well enough to operate - most humans today fail to master one
despite extensive training. There are probably specialists in Starfleet for Vulcan and Klingon medicine, but there's no conceivable way Starfleet could afford to place them aboard the Enterprise
, and no particular motivation to do so.