Heat water and sugar in a medium-size saucepan over medium heat to a simmer, stirring until sugar dissolves, about 4 minutes. Pour sugar mixture into large bowl to cool. Add lime juice.
Working In batches, puree melon in food processor. Stir into sugar syrup. Add liqueur. Pour into shallow metal pan and freeze for 3 hours. Stir mixture every hour, mixing crystals from side of pan into middle.
When mixture is frozen, and working in batches, place granita in food processor. Pulse for 3 to 5 seconds to make a snowlike mixture. Spoon into serving dishes and serve at once. Makes 6 servings.