He didn't say make rings, he said enchant rings. Enchanters can enchant their own rings. There are four BoP enchants for rings: +2 Weapon Damage, +4 All Stats, +20 Healing, and +12 Spell Damage.
I would agree with others, if you're going to drop one profession, I'd say drop skinning. If you choose to pick it up again, it's not too tough to re-level it. LW on the other hand would be a huge chore to re-level.
I would not necessarily recommend enchanting as a replacement for skinning. Enchanting can be costly to level if you don't do it right. There are ways to level it without a big cost, but it can be a money pit. Also, the only thing you'll gain that someone else can't enchant for you are those ring enchants. If you want enchanting for disenchanting to make money with, that's one thing. But, if you just want it to enchant your own gear, I would think hard about picking it up.
I am currently thinking about dropping skinning for mining (to make money), or herbalism (to supply my raiding needs - guildies can make pots for me).
Anyway, I'd say in general a gathering profession is better to drop than a crafting profession.