Funny, just thought of the same thing, so I tried it, since the Watch charger is Qi, but simply a smaller coil (I hear that it might charge non-apple watches).
I tried it with my AirPods case, and it does not work with the Apple Watch charger. It doesn't even recognize it. I also tried it with a non-Apple Watch charger. This charger seems to recognize the case (i.e. the light comes on, and the status says charging), but it does not actually charge, although it gets hot. I suspect that the coil is too small and/or does not provide enough current, I think it's only 350mA.