So I’m not a doctor, but I am an EMT. This looks and sounds like a mild chemical burn. While the band is the obvious culprit, there might be a compounding factor. Things to consider are:
-is wearing an Apple Watch new to you? If not, what other bands have you used? And did they produce a similar result? Is this the first time you’ve used this band?
-do you use lotions? Hand sanitizers? Are they scented?
-do you have any allergies/sensitivities to perfumes or fragrances that could be in soaps or lotions?
-what were you doing prior to noticing the discomfort. Chemical burns can take hours to manifest in a rash if the exposure was mild.
I don’t know for sure how the silicone elastomer bands are manufactured but there is a possibility that there is some residual oil or some other release agent from the molding process left on the surface. A good wash might be all it needs. Or you may just be sensitive to the materials it’s made of. Or the dye used for that particular color.
I say all this, not to overwhelm you, but to get you thinking. I just know from experience that sometimes a chemical burn requires some sleuthing to uncover the triggering factor. Mild ones often are tricky to trace back to the contaminant. Severe ones are frequently obvious because the material that caused it is still evident.
Anyway, hope some or all of that helps.