Watermelon plants often reject the melons. I get that all the time but never looked like that. If that is the side that the WM sat on it might be rotting because of the moisture. It might very well be BER but I never experienced that. BER is usually due to a calcium deficiency. I always add calcium to my WM. I suggest using gypsum instead of lime.
Picked my fist 2 Crimson Sweet WM on Sunday.
Like oneeye mention testing your soil is best but with so many trees, shrubs, vines and veggies it is just to expensive for some us. With time you get to know your deficencies and what to use with what. Next time you grow WM you now know to add calcium.
MOD