Combine 4 cups of the watermelon cubes and 1/2 cup of the water in a blender and puree. Press the mixture through a sieve set over a medium bowl. Pour into a large pitcher. Repeat with the remaining watermelon and water.
Add the sugar and lime juice to the pitcher; stir until the sugar is dissolved.
Pour the aqua fresca into 6 ice-filled glasses. Garnish each drink with a watermelon wedge.