Place the water and sugar into a medium saucepan; stir over heat, without boiling, until sugar dissolves. Bring to the boil; reduce heat to low and simmer, without stirring, for 8-10 minutes or until slightly thickened but not coloured. Remove from the heat and cool to room temperature.
Meanwhile, process honeydew in a food processor until smooth. Strain into a shallow stainless steel bowl. Repeat process with rockmelon and watermelon in clean food processor, straining into separate bowls. Stir one third of the sugar syrup into each bowl.
Cover each fruit mixture with foil and freeze for 3 hours or until almost set. Chop each fruit mixture coarsely with a knife. Process each fruit mixture, separately, with one egg white until smooth. Return each mixture to each bowl.
Cover and freeze for 3 hours or until firm. Serve scoops of sorbets in glass dishes.