You're right - the Islamic material used to support it is in the Hadith, not the Koran. Also (i) the validity of it, i.e. whether or not it really is a statement of Mohammed's views, is strongly disputed and (ii) it could reasonably be interpreted as Mohammed opposing it but not being powerful enough to stop it. Basically, he encountered a woman who cut off parts of girls genitals and told her to cut off less. Assuming it's a genuine hadith, which is hotly disputed.
You can indirectly, but reasonably, use the Koran itself to oppose cutting off bits of a person's genitals, whatever their sex. Altering God's creations is not allowed, simple as that. There's even a passage forbidding making a small cut on a cow's ear (presumably for identification purposes).