Place 2 1/2 cups watermelon and 3/4 cup cold water in blender. Puree until smooth. Pour mix into large pitcher. Repeat 3 more times with remaining watermelon and cold water.Advertisement
Add lime juice, 1/4 cup sugar, and ginger juice to pitcher, and stir to blend. Add more sugar by tablespoonfuls, if desired.
Refrigerate until well chilled, at least 3 hours or up to 8 hours ahead. Remove from refrigerator just before serving. Fill glasses with ice cubes. Fill halfway with juice and finish with sparkling water or club soda. Garnish each glass with lime wedge and serve.