The dish soap didn't "eat his clear coat".... It removed the wax and sealant. Quite likely, the clear coat was already sun damaged, and had a subsequent wax/sealant to make it glossy again. I would bet that another wax/buff would have brought the shine right back.
Don't forget that pool balls are a highly polished phenolic resin. It is perfectly safe to CLEAN (not polish) them with Dawn dish soap, without fear of causing any sort of chemical reaction or material degradation. Though, they will still require polishing, to bring back a high luster shine.
That said, I only reserve the Dawn dish soap and warm water for the absolute grimiest balls, before they go into the polisher. I'm talking sets that have been stored in a barn, or have seen years of nicotine stains. Being in the business, I come across all kinds of nasty stuff... The kind of grime that you aren't going to see from typical play in your house.
Check out the before and after photos of the set below. This set was cleaned with Dawn and warm water, then polished in a Diamond ball polisher, using Chem-Pak Ball Cleaner and Polish.
