Thanks for contacting eXtension. This sounds like two-spotted spider mites. Do you have any pictures? Their webs are very small, and sometimes hard to find. So, I wonder if it may be something else if they are very easy to see. They are usually identified by the white and brown stipling damage on leaves. See some pictures here.
I have tried to transfer pictures, but it is not working.
The day before yesterday, I noticed a long web hanging down from a leaf and removed it with tissue and threw it in the garbage. Yesterday when I went into the greenhouse to water, I found my cantaloupe leaves covered in with webs. My watermelon vines are looking awful now too.
Read this over, and scroll to the bottom of the page and enlarge the images. If this is what your melon plants look like, follow the the options to manage a melon plant infested with spider mites in this WSU Hortsense article: http://hortsense.cahnrs.wsu.edu/Public/FactsheetWeb.aspx?ProblemId=236