We've had a very dry and hot spell but, but it started cooling off pretty fast. From nightly temps in high 80s all summer to now sudden temps in 40s and 50s at night. After the temperature change is when this started showing up. They were watered and looked healthy, aside from some aphid damage, through all the hot weather. The cooler weather also seems to have gotten rid of the aphids.

That looks a little like spider mites. Are there very small insects on the under side? It could just be lack of fertilizer and aging leaves.

You'll either need to submit an improved description or some pictures, if you'd like a good ID. What is the size and shape of the critter? Is it mobile? Where are the 'eggs' located and what is their size and shape?

Thanks for all the suggestions. I did finally get a fungicide / insecticide combination spray. It wasn't organic, although the label states that it's safe to use on fruit and vegetable plants. (I'll have to wash the fruit really well before cutting into it.) The stuff smells horrible, but I think it may have done the trick. There is still leaf damage, but I have a new shoot starting from the main vine and the watermelon fruit doesn't seem to be as soft anymore. Here's for hoping. Next year I'm going to use a fungicide before the plant starts to flower and hopefully that'll prevent this from happening again. Thanks for your help!