I can tell you why you sit in the water. My traditional army doctor told me to do that over 30 years ago when I had chronic kidney problems. It greatly reduces the pain and if the pain is reduced you are more willing to urinate and if you freely urinate you get rid of not only the stones and crystals but also any build up infected urine or material. But you don't have to sit in your own urine you can constantly refill the tub if you want to. Use very warm water, as warm as you can stand it. Not sure why that helps but it does. It does need to cover the lower back to help with all the pain.
As to watermellon I have no idea if that works or not. I'm going to have my daughter do it since I have a feeling she won't take the herbals from Dr Clarks method. I used to use massive amounts of plain cranberry juice and water. Think it took about 4-8 liters for two days to work. And vitamin c in mega doses. Also a little wine can help you relax if your in lots of pain.