First of all I wash underhood all the time with no ill effects. On the newer computer controlled cars you have to worry a little about electrical connectors, but if you just kind of lightly dust it with the car wash wand you will have no trouble.

Secondly, the belt was wet after the wash causing it to slip. This caused the compressor rotation sensor to cause the compressor to kick out. It will probably be okay today. No worries.

Good luck,