Yes, a optometrist can diagnose pink eyes. Actually most optometrist are also able to diagnose and treat some common eye problems. They have such kind of practice before their career. They can diagnose red eyes, dry eyes, pink eyes and so on. So, if you feel discomfort in eyes and want to have a eye check, just go ahead!
Your eyes appear red and itches, it may be caused by pink eyes with bacterial or it is just the symptom of dry eyes that caused by eyes fatigue. There are many reasons that would lead to your trouble. Pink eye is the common name of the disease infectious conjunctivitis. It is an acute infectious eye inflammation which possibly infected through articles contacts of patients. Actually, it's easy to get but still have way to protect, good health habit would prevent the bacteria or viruses pass through hands to eyes. I suggest you go to hospital to find a doctor to deal with your disease but not go to optometrist to diagnose it. As the optometrist's responsibility is to measure the degree of the eyes prescription of myopia.