Put 4 cups watermelon in a blender; process until smooth. Strain through a sieve into a pitcher; discard solids. Repeat procedure with remaining watermelon cubes. Combine lime juice, sugar, and ginger in a bowl; stir with a whisk until sugar dissolves. Add lime juice mixture and watermelon balls to watermelon mixture; stir in sparkling water just before serving. Serve over cracked ice, if desired.