Scoop the fruit into a blender, then add the water, sugar and lime. Blend until smooth. Add more water, if necessary, to achieve a light, easy-to-drink consistency. Taste and season with more sugar and lime if you think necessary. Refrigerate to chill or serve over ice.
You can add extra flavorings (mint to honeydew, orange juice to cantaloupe, and blended raspberries to watermelon) if you like.