Listen to @rayrose . Ray knows Watermelons. (to borrow the famous Nike slogan. ha) I've tried potted WMs myself more than once and the results have ALWAYS been just exactly what he said....lots of beautiful vines, but the melons never get very large. Even when I've used very large pots (15 gallons) and kept them well watered and fertilized, it just doesn't work. I have gotten some (round ones) to be a little bigger than a baseball, but then that's it....they neither die or grow, just sit there the rest of the summer. So, @UrbanAggie , feel free to experiment if you wish- its always fun to try new things- but I think you're potential for success is pretty low..
@UnderDawgAl I'm really enjoying watching your experiences and excitement! And you are right....when conditions are good, watermelons grow at an astounding rate. I had a volunteer come up on edge of my patio this year (right where I spit my seeds last year, of course!) and I've turned the vines so they are growing over my concrete patio. Even after a lifetime of growing watermelons, I'm astounded by how much those vines grow every single day. It also still amazes me just how much bio-mass (vines, melons, etc) is created from a single watermelon plant. It has already completely covered an area about 7 ft x 8 ft is still growing like crazy. By that way, be really careful when you are doing all this weighing and holding the melons for photos. Its hard to explain why, but sometimes handling them will cause them to abort. Certainly not always and I've handled mine a fair bit with no problems. But I've also had some abort from what I'm pretty sure was from being handled too much. So just to be on the safe side since you don't have many melons, I recommend that you handle them as little as you can.
Good luck and keep having fun with watermelons.....I think they are the funnest thing I've ever grown and they have a high rate of return for a small investment of time, effort, and money. Yea Watermelons!