Yeah, I’ve been wearing an Apple Watch for years. It’s funny how people make these broad obviously silly statements like “lights don’t burn” because there are any number of examples of light sources that can burn people, including lasers which can down aircraft. Those however are orders of magnitude more powerful than anything that can fit in a watch.1. All sorts of wrong. Powerful lasers can punch holes through metal. Buy a PAR64 lamp (not LED) and turn it on and let's see how long you can keep your hand on top of it.
2. Agreed, not enough amperage to do anything, nor are you touching any electrical contacts anywhere on the watch.
My guess is it is an allergic skin reaction to the watch band or whatever the watch is made of or you have some foreign substance on the watch that is causing it.
I think the OP is obviously having some kind of reaction, but it’s not outside the realm of possibility that the watch has some kind of defect, perhaps with the battery, that is causing it to overheat. Easily verifiable: is the watch hot to the touch? If not, it’s probably not burning you. If it is, go directly back to where you got it immediately and return it, lithium battery fires are no joke. I can’t imagine a defect that would cause the light sources to “pew pew!” Laser blast the skin.