Taxidermy.

Do you cut the tips of their claws? Taking off just the very pointy tip with nail clippers or fancy cat grooming clippers once a week can help. Sneak up on them when they're asleep and cut a few toes at a time if they don't like it. If his claws aren't as pointy they won't do as much damage, and won't get stuck in the screen holes.

And what's your scratching post situation?

I use a post made of two 2x4s nailed together and wrapped in thick rope, held with staples. You can buy scratching posts like that too. I don't like the ones with carpet. But cats have to have something to scratch.

You can spray bitter apple on the screens to see if it cuts down the licking.

In general, the cat might be bored. Might need to explore the whole genre of cat exercise and entertainment options.

Well, I start in kittenhood, teaching them that claws never touch screens. That way, I can allow them out onto my screened-in deck, and not worry so much about them accidentally clawing a hole in the screen and getting out, or pushing out a window screen.

But with an adult cat, or a slow learner, you either need more hard-core screens or a barrier. Or both. The metal screens are very nearly claw-proof, but a large and heavy cat can still force the screen out the window if they leap on it. So what some people do is to install shutters on the INSIDES of the windows, or put lattice over the screens on the outside to reinforce the screens.

This is useful for keeping dogs and toddlers from falling out of windows, too, not just cats.

Keep his toes trimmed, get heavy duty screens, and if he's still determined to climb them, try double-sided tape on the side that the cats hangs from. Cats don't like sticky feet and he'll quickly learn to not jump/stick to the screens.

Get the metal netting screens, rather than fiberglass netting.

Or use plexiglass on the bottom part of the inside of the screen (well: if it's a very tall window- I've done this with screen doors and it saves a bundle).

Well you could get his front claws out. Like take him to a groomer/veternairan and have (JUST) his front claws out. So it won't happen ever again. And if he claws you then it won't happen either.