Another photo

I see one tiny watermelon plant. It seems stunted, the leaves are usually much larger. The plant doesn't seem to have an optium growing environment. Too crowded with other plants. I hate to say it, but I wouldn't expect much from it.

Can I transplant it? Does it need to grow vertical?

That watermelon does looks a bit scrawny. It wouldn't be worth the effort to transplant it. Cucurbits don't like their roots to be disturbed. You can replant new seeds in a different spot of your garden if you are willing to try. However, watermelon plants get huge, much bigger than cucumbers. So if you are having trouble controlling your cukes you are going to have a devil of a time controlling the watermelons. (I saw your other post). As for growing vertical, watermelons are best left to sprawl along the ground.

Rodney