All four of them are grammatically and semantically valid. All are natural, and all have different meanings or nuance - albeit slightly in some cases.
Let's look at the third first. In that case, we have an active voice verb in the final clause, to hurt. It is intransitive, and your stomach is the subject. When to hurt is intransitive, it generally means that the subject is the site of a sensation of pain, or if the subject is a being, that the being experiences pain. So, this says that you will experience pain in your stomach if you use that supplement.
The other three have your stomach as the subject of the verb in the final sentence, but always in the passive voice. That means they would be the object in the active voice, but without a specified subject. So it is effectively saying that some unspecified something will either damage, harm, or hurt your stomach.
Where anything does any of those things to a living thing, or part of a living thing, or sometimes to any physical object, the meaning overlaps. If something is harmed, it is in some way damaged, and if something is hurt there's arguably a degree of harm. Hurt is the weakest, and most informal. Damage, to me, gives the strongest sense of real, lasting harm/damage, while harm is a little milder. That may vary between dialects, though.
I would say damage seems the most natural, if you're intending to give that sense at all.