Yeah, those are not watermelon leaves. To me the leaves could be anything from squash, cantaloupe, cucumber, or gourd leaves. They look small for pumpkins Im used to. I dont see a lot of tendrils holding those vines to the fence. It looks more like they have been woven into the fence.
My first guess would be some type of melon. There are a whole lot in that family other than watermelon and cantaloupe. I really dont think that fruit is cucumber. Some type of winter squash or gourd wouldnt shock me, but that just looks like a melon of some type to me.
Have you considered splitting one open to see what it looks like inside?
after looking over what Johnny's has on their site i'm wondering if they could have grown any of their armenian cukes too close to any melons and they cross pollinated. armenian cukes are closer related to melons and i think i've heard those can cross easily. i'm thinking its not going to be a watermelon cross after all. i hope whatever it is it will be sweet tasting for you!