Gradually add watermelon to container of electric blender, processing until smooth. Combine 10 1/2 cups watermelon mixture and sugar in a large bowl; freeze until slushy. Beat well.
Beat egg whites (at room temperature) until stiff peaks form; fold into watermelon mixture.
Pour mixture into freezer can of a 1-gallon hand-turned or electric freezer. Freeze according to manufacturer's instructions.
Note: Ice may be stored in freezer compartment of refrigerator until serving time. Let stand at room temperature 10 to 15 minutes before serving.