A "pure" competitive market would, but we don't have that and we're not going to get it. Eliminate insurance "networks", force hospitals to publish price lists and give the same price to every patient, make the medical statistics of every hospital and provider publicly accessible, and costs will plummet. As of now we have a hybrid system that combines the worst of both methods.
I think that's the key point. That would give insurances much more leverage to negotiate lower prices for everyone. Even other smaller insurances would profit... which would increase competition between the insurances... which would lower insurance costs for the people.