I have never had a watermelon ripen after picking. You can use panty hose as a support but you would need the plus size.

I actually did not intend to hang the watermelons, I am planning on having them ripen on top of the arbor. The vine has already grown to the top and watermelons are about an inch round. I picked off all the flowers and everything as it made it's way to the top of the arbor so it actually is perfect for them to just lay on top. I just didn't know if I had to turn them over, somebody told that. Also somebody told me they would ripen more evenly if they were hanging.

Smaller watermelons could ripen more evenly but not the 10-20+ pounders you may get if you have a good season. That weight would probably pull the vines down. But since this is an experiment try it. You could put a open weave container under some of the melons. The weight of melons will sometimes cause the melon to go out of round which is another reason for the smaller one.

At one older conservatory, Longview I think, someone had made small hammocks to hold the fruit that dangled below the trellis.

I guess I shouldn't call it a trellis because it's an arbor, sorry. I want to lay the melons on Top of the arbor, not hang. Since my arbor is not flat on top I suppose the melons will take on a weird shape but will that affect taste? I will probably try and "hammock" a few just to see if I'm able.