1. Combine the water and sugar in a small saucepan. Bring to a boil over medium heat, stirring to dissolve the sugar. Reduce heat and simmer, uncovered, for 5 minutes. Cool to room temperature. Stir in finely shredded peel.
2. Place papaya chunks in a blender container or food processor bowl; cover and blend or process until smooth. (If necessary add a little of the fruit juice to help mixture process.) Stir together papaya, fruit juice, and the cooled sugar syrup.
3. Pour mixture into a small nonmetal freezer container; cover. Freeze at least 4 hours or until firm. Break the mixture into small chunks; transfer to a medium chilled bowl. Beat with an electric mixer until slushy. Return mixture to pan. Cover; freeze at least 4 hours more or overnight.
4. To serve, let mixture stand at room temperature for 10 minutes. Scrape a large spoon or ice cream scoop across surface of frozen mixture.