What I do on sport climbs is take my watch off and put it in my pack top pocket. You’ll only be away from the watch for 10-15 min at most so it’s easy to refer to it occasionally. On longer rock climbs I attach/buckle the watch wrist strap snuggly around the waist belt of my harness close to the belay loop. It is very unlikely to contact the rock there (min never has). It’s out of the way and I can refer to it from time to time to make sure we’re not going to get benighted on the climb 🙂
Scott