I looked on the wiki to find out how but didn't see anything.

Craig Boone got his leg injured in a fight, and now he's limping around everywhere, whining about it. I didn't see any option in his menu or dialogue to heal limbs, and me sleeping doesn't heal him either (makes sense, lol).

Do I have to use one of my limb healing items on him, or can I pay a doctor to heal him? I don't have many limb healing items, and would rather save them for myself.

In the companion "wheel" there is the ability to use a stimpack which I believe will heal him. It's around 1-2:00 on the wheel.

The companions health completely resets after a fight, and the limbs are restored (it's scripted). But if they are injured out of combat they don't heal. If he is sill injured after the fight he probably was injured OUT of the fight itself (walking on a mine or eing hit by an explosion or trap or something)
Then to heal your Boone you mus lead him in a fight. Any fight will do and he will be automatically healed, even if he "dies" (unless you play in hardcoe,If so turn of the hardcoe mode before the fight)
If he doesn't heal after the fight then you have a serious bug meaning his script doesn't run anymore. He could die for good or do other silly things and you'd better reload a previous game.

There is no way to have him use a medical bag or to have a doctor heal him. You can only give him stimpaks but the simpak will not heal the limbs on a companion, only on the player.
In last resort you can use the console command to heal him manually (resethealth)

Thanks, I totally forgot Stimpacks heal crippled limbs too. Those Hydras and Doctor's Bags just got less valuable.

I was wondering why I couldn't heal the limb with a Stimpack from the wheel, but I see now it's because it won't let you unless they're missing HP, even if they're crippled. I just shot him in the head, then used one on him. He didn't seem to mind.