Multiple potential complications, having grown in Georgia clay:
#1: calcium: As mentioned above, you might have a calcium deficiency of some sort. You have a couple of options: get a good mineral supplement, or grow plants that "mine for calcium" such as dandelion, then using the plants as fertilizer/mulch (calcium isn't available until the plant decomposes).
#2: Soil texture: If it's truly HEAVY clay, you're going to need to shovel in some organic matter to loosen it up and as a result improve the cation-exchange capacity, as mentioned above.
#3: water retention: watermelons don't like too much water, or soil that doesn't drain. If you're having drainage issues, I suggest mixing a bunch of compost in with native soil, build it up into a hill, and plant in that hill, so the water doesn't puddle and weaken the plant to infection.
#4: soil pH: in Georgia, we tend to have acidic soils. If the soil gets too acidic, it can hinder the absorption of calcium. Get your soil pH tested, and add limestone or ashes as needed.