well, im not an expert on watermelons, but i do remember hearing that if the leaves curl up on tomato, iit from over-watering. -(or not enough)

also kinda looks like a mildew,r virus or fungus on the leaves which also could be due to over-watering . and if you can water the roots, and not get it on the leaves.
(unless you are doing a foliar spray, which may help)

more info would help
like Temps, how much water it gets, how big it is, how much fertilizer and what kind you have fed it, how much sun...
any fruit on the plant ?

link below has some good info

Here is a link that might be useful: http://forums.gardenweb.com/forums/load/okgard/msg0715504313394.html

Thank you for your reply. Just read the link you provided. It very well could be too much water. I use a sprinkler to water these and other plants in my raised beds. So water does get on the leaves.

No fruit yet on the plants--they were transplanted at the beginning of this month. (I'm in zone 6b.) It's been hot--in the 80s and 90s. All-day sun. Have not fertilized at all, but the soil in the beds is full of compost and nutrient-rich.

I will detach the sprinkler and water everything with a sprayer attachment, setting it on soaker.

Is it too late for the watermelons to recover?

Another photo. I pulled this plant out of the ground. Is this mosaic virus? If so, I guess all of my plants must be destroyed. Is there any way to sterilize the soil? Must I completely empty my raised bed?

It looks like some kind of fungus. You might try spraying with a good fungicide, and eliminate the over head watering. That should solve the problem.

Thank you, rayrose!

You would have to show an extremely close-up view for a diagnosis of mosaic virus to be accurate.

Many squash, melon, pumpkin leaves are prone to mildew. In a continuously wet environment, it is no problem. Where there are wettings and dryings at certain temperatures, it takes off.

Any herbicides used anywhere in the vicinity?

Not in my yard, at least, but I don't know about the neighbors. Every other plant looks fine.