Your stomach is filled with diluted hydrochloric acid, with a pH somewhere between 1 and 3. This is strong enough to denature proteins on contact.
A seed from a "seedless" watermelon would be destroyed in seconds. However, the seed coat on a regular water melon seed is designed to dissolve while in the stomach, readying the seed for germination when it is excreted from your body.
The seed will leave your digestive tract somewhere between 24 and 72 hours after you swallowed it, which is far too little time for the seed to germinate. Even if you had a medical condition that slowed your digestive tract, the seed wouldn't receive the light and water needed for it to grow into a watermelon.
This doesn't mean you're entirely safe from plants growing inside your body. If you eat Skittles, it's entirely possible a Skittle tree will grow out of your stomach. :)
Posted 1972 day ago