Can any harm come of eating watermelon seeds?

Um, no.

You can even buy them.

The only common potentially dangerous seeds I know of belong to almonds, apples, apricots, peaches, plums, cherries, and other stone fruits. These contain a cyanide and sugar compound known as amygdalin. When metabolized it breaks down into hydrogen cyanide (HCN). In all cases the toxin is inside the seeds and will not be exposed to the body unless the seeds are chewed. Of these, the only seed commonly consumed would be the apple seed, usually inadvertently, or by daring children. It would take a large amount (can't find a reliable reference - one source said 1 cup) of well chewed seeds to poison you. Regarding almonds, only the bitter almonds have cyanide levels to be concerned about (we eat sweet almonds).

So, besides the apple, these all belong to the Prunus genus of plants. Cyanide is just their thing. Watermelon belongs to the Citrullus genus.

There's a fringe group of naturalists/alternative-medicine practitioners who swear that the poison Hobodave mentioned also cures cancer and has been suppressed by the FDA and pharmaceutical companies in a big conspiracy. I'll let you decide...

A better question: Aren't most of your watermelons seedless by now? I haven't eaten a "seeded" melon for a few years at least.

I bought white watermelon seeds which were supposed to be eaten as you would pumpkin seeds. I chewed everything well. They were delicious.

However, a few hours later, I became ill; vomiting with diarrhea. I did not have a fever or aches & pains as with the flu. I do believe it was from the watermelon seeds.

Fortunately, I was fine the next day.