Make a simple syrup by mixing the sugar with 3/4 cup water until the sugar dissolves.
Put the melon in a blender with the lime juice and blend until smooth. Add about 1 cup of the simple syrup to the blender and blend. Taste and add more simple syrup if needed. Refrigerated until chilled.
Add some ice to each wine glass. Add 1/3 cup melon puree to each and top with sparkling wine. Garnish with a piece of melon and slice of lime on the rim of each glass.