If an uncut watermelon needs to ripen, store at room temperature. Watermelon that has ripened but has not been cut should be refrigerated. Uncut, ripe watermelon keeps longer and is more refreshing to eat when chilled. Watermelon that has been cut should be wrapped tightly in plastic wrap or placed in an airtight container. This will keep the cut watermelon from absorbing the odors of other foods and prevent it from drying out. Eat as soon as possible.