I grow watermelons vertically; they are sugar baby, so probably smaller than what you are growing (6-8 lbs), but it's been a good experience for me. Once they start to get large, you can give them support with a sling or hammock made from stockings tied to your trellis or a strong structure. This makes sure they don't fall off the vine =) Here's some more information from NM State University:

Here is a link that might be useful: Vertical garden watermelon on trellis