I thought I'd post a follow-up about what doesn't work when you accidentally get bug spray on your watch.
I tried using soap, listerine, alcohol, hydrogen peroxide, saline solution, and WD-40, but the watch surface is still not clear like it was before. I don't think the screen is damaged, but the bug spray on it won't come off. It's more a nuisance than a big deal. I can see the time, but also some sprinkly glare.
Moral of the story: apply bug spray to body, not watch.