The day of serving, cut flesh out of remaining half of the watermelon. Puree in a food processor fitted with a knife blade. Strain pulp through a medium-mesh strainer set over a large bowl; discard solids. (Alternatively, cut watermelon into 1-inch pieces and place in a large bowl. Mash melon with a fork, then strain.)
Remove balls from the refrigerator; strain unabsorbed vodka into the fresh watermelon juice. String 3 to 4 of the drained balls onto each of 4 cocktail picks. Divide remaining balls among 4 chilled highball glasses. Top with watermelon juice, adding more vodka if desired. Garnish with the cocktail picks.