Are there any natural supplements that can help heal conjunctivitis?

Not many. Most forms of conjunctivitis are caused by inflammation or infection. Infections should be treated with the appropriate medications (antibiotics, antivirals), and there are unfortunately no known supplements that will help. Allergic conjunctivitis does not respond well to supplements but as long as they do not worsen the condition, they certainly can be tried.
Fish oil? Depends upon the cause. Most commonly, dry eye patients take omega-3 fatty acid supplements to help with the clinical symptoms of dry eye.