In food processor, place watermelon. Cover; process until smooth. Into large pitcher, strain watermelon puree; discard solids. Add sparkling water, rum and limeade concentrate; stir until concentrate is dissolved.
In each of 4 tall straight glasses, muddle 1 tablespoon mint leaves in bottom of glass. Add crushed ice to glasses; pour watermelon mixture over ice. Garnish with lime wedges.