I've never heard of avoiding bananas and watermelon during pregnancy, I'd be inclined to say it's just a myth. A midwife at my local hospital was absolutely horrified when she found out I was eating peanut butter on toast while pregnant with DD though, she told me I had to avoid peanuts and strawberries because it could cause an allergic reaction in the fetus and I could miscarry or have a deformed baby or something... I totally ignored her because I could find absolutely no other evidence from any source that eating peanuts and strawberries while pregnant could kill or harm the baby in any way...
So I'd put them both in the myth basket and ENJOY
(I hate bananas though, so you're quite welcome to them haha )