I tend to discount the scene at the end a bit, based on the dialog SalvorHardin
graciously posted. Still, I think something's up.
Either Bones is really worried about the psychological effects that everything that's been going on have had on him (which is certainly possible, even though he actually does
examine him) or there's something else going on.
I don't recall any health concerns for Kirk in the Primeverse. Orion gonorrhea, maybe.
What's a bit more interesting, though, is this: if he does have some sort of terminal or even potentially fatal illness, then that explains why he was so ready to let himself die at the end of the film. Indeed, it might help explain his earlier recklessness too.