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I planted both near water and then went spelunking for a few real-time hours. I came back and they both had not yet grown more than the first stage past the initial stage when they look like tiny green leaves in the soil. Right now, both the pumpkins and the melons plots look like light brown stalks sticking up out of the ground and don't seem to have changed any for a long time.
I don't really grow melons and pumpkins but I get that with wheat sometimes, I plant a few rows and they grow steadily but one or two patches are still on the first stage of growing.
You could speed things up much faster by killing a skeleton and using the bones to make bone meal, bone meal makes plants grow instantly but with melons and pumpkins it grows the stem completely but you still have to wait for the fruit to grow.
Thanks to all of you for the help. The first big help was knowing that I have to leave a spot for the grown melon to plop down. Initially I had surrounded the plots with wood slabs. Mistake!
I knew about the torch light so much obliged. I also created a small water channel that runs right down the middle of the two rows I created. They started popping in within a minute after I removed the wood slabs so all is good now. :)