Pour watermelon and pomegranate into the blender vase. Puree in the blender. Sieve to get rid of solids, pressing with a spoon to squeeze as much liquid as possible.
Mix the watermelon/pomegranate juice with the lime juice and add sugar (or sweetener) to taste.
Pour equal portions into 6 highball glasses. Add equal amounts of ice.
Pour equal amounts of carbonated water into each glass and serve immediately (before water goes completely flat).
Notes
It's OK that once you add the sparkling water it is less sweet than you originally intended. It's not supposed to be too sweet. Let the sparkling water, well, sparkle.