A broken blood vessel heals on its own so there is nothing much you can do. The conjunctiva, the thin membrane covering the front part of your eye, contains tiny blood vessels. When these blood vessels break, a bright red spot appears on the sclera, the thick white outer layer of your eye. Although the broken blood vessels may look dramatic and alarming, they are harmless and will go away on their own in about 5-10 days when the blood gets reabsorbed.