I'm a medical historian and I think about this all the time - it's a more complex question than it initially appears to be. The typical line of reasoning is to assume that medical treatments "progress" through history as the efficacious ones win adherents and the ones that don't work fall out of favor. The problem with this is that the most popular medical treatment in the European and Islamic world for 2,000 years (bleeding) was medically worthless (well, unless you happen to have very high blood pressure).
So were all premodern doctors idiots who couldn't practice empiricism? Well, no - many of them are extremely intelligent, inquisitive people (people like Isaac Newton and Robert Boyle accepted bleeding as a treatment, so clearly "they weren't smart" or "they didn't understand the scientific method" isn't a good explanation). I think the answer is two pronged. One is simply that the placebo effect is extremely powerful, so even cures that we consider not to be efficacious in a clinical setting still "worked" in daily practice as placebos. The second is to look more closely at what someone with the scientific mind of Robert Boyle made of bleeding. He actually did do quasi-experimental trials involving blood loss and concluded that "excessive" bleeding was bad for human health. By the same token, he started doing chemical assays of various mineral and animal-based cures (like eating burnt lizards for epilepsy, and things like that) and began to lay the foundations for our later (18th/19th century) understanding of which minerals and vitamins are essential for human health. It's entirely possible that Greco-Roman doctors were doing similar things, but Boyle had the benefit of the ready dissemination of his findings via print and the world's first scientific journal (Philosophical Transactions), which happened to be edited by his close associate (Henry Oldenberg).
In short I think there was a kind of practical empiricism at work in the history of medicine that has always existed. But that it didn't begin to coalesce into clinical effective treatments until print culture made it possible for people to share their findings on a global level. This wasn't a simple march out of ignorance toward truth, though. And the reason for this is that the placebo effect of remedies like bleeding often "worked" just as well as the partially-effective treatments of premodern times, like, for instance, supplementing poor diets by eating the ashes of various animal and plant products.