Those who have a severe enough allergy to watermelon will most likely have an adverse reaction to eating it, which can include nausea, bloating, and diarrhea. Although this reaction is fairly rare, it does happen from time to time in those who experience an allergic reaction.
Taking into account that watermelon is about 92% water, it is not surprising that this food would have this effect on your bowels. While there are some people who are extremely allergic to this food, others experience no adverse reaction whatsoever.
If you do experience any diarrhea from eating watermelon, it is important for you to stay as hydrated as possible until it passes completely.