Peroxide is not what heals your cut. Your body is what heals your cut. Peroxide can help clean it up though and make it heal faster.

Do you know why peroxide stings when you put it on a wound? It's because it is killing your live skin cells. If it just killed the germs in the wound, you would not feel it since the germs are not tied into your nervous system. Peroxide can be used if you have nothing else, but you are much better off using an antiseptic like Bactine (my favorite as it will run deeper into a wound without having to force it in like you do with Neosporin) and a clean bandage every day. Make sure that air can flow through the bandage to keep the wound from going septic, a condition caused when anaerobic bacteria take over.

Peroxide is stored in vesicles called peroxisomes that help to kill living matter no longer required by the body (such as superfluous or old cells) or to detroy invading pathogens by ingesting them and releasing the vesicle contents onto them. So the peroxide will help to sterilise, certainly, but will also damage the fibrin meshwork and clot aggregation that helped to stem the flow of blood, as well as damaging and possibly killing local cells. It will also probably sting like a b!tch. So I suppose you could use it to sterilise if nothing better is at hand, but it's kind of like using bleach to sterilise, it will cause harm as well as good. If you can get hold of a betadine wash or something similar, that would be better. If not, as a quick home remedy, you could try boiling salt water, but this hurts like hell and isn't as effective. Eventually the cut will heal on it's own anyway, just remember to keep it clean and replace the dressing every day or two (try to keep it dry as well). If it's a wide or deep cut, try using steristrips across it to form a bridge between both sides (you can maybe get these from chemists) to seal the wound better so that it isn't too exposed.