Weak hydrogen peroxide (1% or less) is quite a good mouthwash and gargle, because as Anders said it kills bacteria. Swallowing small amounts of this concentration would probably do no harm as it would almost certainly dissociate into water and oxygen on contact with the gunk in the stomach, resulting in nothing worse than an oxygen burp.
I don't know of any way it could get into the body as a whole from the stomach, even if it could cure anything if it got there. Since peroxide radicals are generally regarded as a bad thing (they are the sorts of things antioxidants clear up), I can't see how it would do anything but harm if it could get into the body.
I have a friend who swears his prostate cancer was partly cured by drinking a mixture of H2O2 and Vitamin C. That's right .. an oxidizing agent mixed with an antioxidant. Nothing will convince him that this (and other alternative medicine for which he paid a fortune) probably had nothing to do with what was (we hope) a spontaneous remission.
H2O2 sold in chemist's shops for hair bleaching etc., is usually rated as "10 volumes" (3%) or "20 volumes" (6%), so needs diluting by a factor of at least 3 or 6 respectively for use as a mouthwash. This stuff often contains some phosphoric acid, which is probably not great to rinse with but it's in Coca Cola so it can't be too dangerous.
Stronger H2O2, particularly the 35% stuff sold online as "food grade" is harmful to touch and very dangerous to swallow or even use as a mouthwash. I wonder how many people who have bought this have not realised it should be diluted by at least 1 part in 34 parts of water.
Incidentally, "food grade" does not mean it is safe for consumption (diluted or otherwise), it just means it's OK to use for cleaning food preparation surfaces such as kitchen worktops (but wear heavy duty rubber gloves!).