Pink eye in goats is caused by infectious and contagious bacterial disease. It can be usually treated with any number of antibiotics that are injected or placed directly on the eye.
Pink eye is unlikely to heal on its own, and is extremely contagious. Your only real alternative is to take your goats to a veterinarian. He will give you a prescription which will cure it.
Treatment for pink eye in goats (and cattle) are different from other species. You must get Gentocin spray which can be prescribed from a vet. It is possible to just let it run its course, but this may cause other goats to get it too.
