seek medical advice.
Now, you may just need to build up your wrists. You can do this with some simple exercises:
1. grab a barbell. Kneel on the floor, with your forearms resting on a bench, barbell gripped firmly in both paws. Relax your grip until the barbell rolls down your fingers. Roll it back up by making a fist. Curl your fists in towards your forearms. You have now completed the first of a set of forearm curls. Repeat as necessary.
1a. if no bench nearby, do the same movement, but with your arms hanging loosely in front of you, knuckles touching your thighs.
2. find a 10 lb plate (or heavy tome, as you are a bookworm). Grip it with the tips of your fingers, and hold it out that way. Raise it and lower it. Rinse and repeat.
3. perform the standard push-up with the following variation - do 'em on your fingertips.
4. grab a dumbbell (no, not me!). Hold it in a hammer-like grip, with the plates on top of and below your fist. Now, make a movement as though you were hammering in a nail.
These should help strengthen the itsy-bitsy muscles in your forearm.
It may also be something as simple, as changing your hand position while performing push-ups for a while. Space 'em out a bit. Bring 'em in a bit. Move 'em forward. Move 'em back. See if you can find a position that doesn't hurt.
And, yeah, try doin' 'em on your knuckles. Make fists and push off the top two knuckles.
But, first, seek medical advice.