This review by Katerina Ierodiakonou of Philip J. van der Eijk's Medicine and Philosophy in Classical Antiquity: Doctors and Philosophers on Nature, Soul, Health and Disease is interesting. She says:

....as van der Eijk points out, what used to characterise the earlier study of ancient medicine, and in general the study of ancient thought, was some kind of teleological progressivism that paid particular attention to those aspects in which classical medicine was regarded as having managed to solve problems which modern medicine considers as central, or at least to suggest the beginnings of solutions to these problems. That is to say, it used to be the case that those who studied ancient medicine had the tendency to isolate from their ancient context those elements which they regarded interesting from their modern perspective, and to seek in the ancient texts possible answers to questions which arose out of the state of medicine or the culture of their times. It is as if they needed to appropriate the works of ancient medical doctors, in order to be able to learn something from them.

Studying the ancient texts in their context helps us to make better sense of the theories found in them, theories which otherwise often sound naïve. We also avoid oversimplifications and become more sensitive to the subtle differences between ancient views....This, briefly, is the new approach to ancient medicine which van der Eijk advocates. And his inquiry into the interrelations between ancient medicine and philosophy comes to cover at least some of the gaps which were left by the earlier approach.

Alas we are not given much insight. Pity. Because it is here that we would find a conception of medicine and health as being embedded both in science and in the humanities. In treating disease and promoting health, medicine involves presuppositions about human goals and values, eg., those relating to wellbeing, treating patients as objects rather than subjects, or the criteria for health.

My understanding is that in classical Greece philosophy and medicine were closely linked as both professions continued to borrow ideas and methodology from each other. So there a reciprocal influence between philosophy and medicine. My memory of reading classic philsoophy is that naturalist philosophers like Anaxagoras and Protagoras questioned the role of the gods in the universe and that the doctors sought a rational rather than divine cause for disease.

This text examines a few of the early examples of Hippocratic writings, dating mostly from the end of the fifth century B.C. It says that their struggle:

... to apply or criticise philosophical concepts and methodology is typical of their rationalist understanding of human nature, whereby explanations of the cause of disease and health are given in non-theological terms as well as in a consistent, logical fashion. However, just as much as these works were influenced by philosophical thought, they to no less extent affected philosophical writers like Plato and Aristotle, as well as their more contemporaneous Sophists. Although some of the authors may have been disgruntled, the intertwining nexus between philosophical and medical writing was impossible to cut prior to modern times.