Yep, you definitely can
There's this one episode of Rugrats, that old school Nickelodeon cartoon about babies who go on adventures, that's stuck with me over the years. Chuckie Finster, resident redhead and scaredy cat, eats a watermelon seed accidentally and then the babies shrink themselves so they can retrieve the seed and prevent a watermelon from growing inside of Chuckie's belly and causing him to explode. Now, I know that you can't shrink your friends so they can save you from a seed, but what does happen when you eat watermelon seeds? Can you eat watermelon seeds safely? The short answer is yes, and you don't even have to worry that a watermelon will start growing in your stomach.
You can eat watermelon seeds raw, straight from the fruit. They're nontoxic, and the seeds won't be able to grow in your stomach acid. (Promise.) But that's far from the best way to eat watermelon seeds. If anything raw watermelon seeds, though edible, are more of a bother than an actually delicious snack. But you don't have to do much to make watermelon seeds totally delicious.
The easiest way to enjoy watermelon seeds is to roast them. And all you really need to make roasted watermelon seeds at home is an oven, a bit of salt, and some olive oil. After rinsing and drying the seeds, toss in olive oil and salt, then place on a baking sheet. Bake them at 325°F for about 10 to 15 minutes. To eat, simply crack the seeds off and snack—as you would with sunflower seeds.
If you're willing to put in a little more effort, you can make sprouted watermelon seeds. Sprouted seeds are seeds that have started to germinate and grow, and people claim they've got some additional health and nutritional benefits over unsprouted seeds. All you have to make sprouted watermelon seeds is to let the seeds soak in water for about a week, until they start growing. They're great on their own, as a snack, but they're even better when added to your morning bowl of granola with yogurt or even cereal. And if making your own sprouted watermelon seeds still intimidates you, you can buy sprouted watermelon seeds online or at health food stores.
So go ahead, and eat watermelon seeds with abandon.