I did not have any "fine" sugar hanging around in my sparse kitchen cabinets, so instead, I took 1 cup of granulated sugar and ran it through a food processor. Worked like a charm. I am sure you can tell the significant difference between picture "1" and picture "2" above and the sugar granular size, right?
In a medium pan, bring the heat up to a low setting and add in the processed granulated sugar and water. Let the sugar dissolve into the pan for a few minutes, and then bring the water up to a boil for about another 5 minutes. Pull from heat and set aside for roughly a half hour.
While the syrupy mixture cools, take the chopped up watermelon sans rind and toss it into a blender. Puree that little bugger until it is nice and smooth.
Once the syrup mixture cools, add in the watermelon juice and stir it up. Pour into a Tupperware container with a cover and put in the freezer for about 4 hours, or until it is sitting in there on its own. (I started at 7pm)
After 4 hours, pull out and use a metal spoon to mush up the watermelon/juice mixture a bit before throwing it into a food processor. Pulse until it is smooth and then put back into the Tupperware to freeze for another 4 hours. (finished this at 11pm)
Repeat the last step -- pull out of freezer, us a metal spoon to mush up the watermelon/juice mixture, and throw in the food processor. Pulse until smooth and put back in the freezer for another 4 hours. (for some unknown reason I actually woke up at 4am and decide this was the perfect opportunity to get a head start on this step--I am such a dork.)
Take the watermelon/juice mixture out and throw it into the food processor once again, but this time add in the egg whites. Pulse until smooth and throw back into the Tupperware to go in the freezer for another 4 hours (after getting up in the middle of the night, I actually woke up at 8am to do this step without setting the alarm clock)
Finally, take this little bugger out of the freezer and scoop into a dish to enjoy happily. Simply perfect for a summer day.