What is the etymology of 'physician'?

Perhaps the easiest way to understand the emergence of physician is by looking at the allied term physic, which Merriam Webster's Eleventh Collegiate Dictionary (2003) defines and derives as follows:

physic n [ME physik natural science, art of medicine, fr. AF phisique, fisik, fr. L physica, sing. natural science, fr. Gk physikē, fr. fem. of physikos—more at PHYSICS] (14c) 1 a : the art or practice of healing disease b : the practice or profession of medicine 2 : a medicinal agent or or preparation; esp : PURGATIVE 3 : archaic : NATURAL SCIENCE

So a physician (derived from "ME phisicien, fisicien, fr. AF, fr. phisique medicine" according to MW) was simply someone who practiced physic—a natural science of healing. The only complication to this point is that MW says that physician emerged in the thirteenth century—one century before physic did.

As for the derivation of physics, MW has this:

[L physica, pl., natural science, fr. Gk physika, fr. neut. pl. of physikos of nature, fr. physis growth, nature, fr. phyein, to bring forth—more at BE] (1715)

The word physicist, as Rossitten notes in another answer, came along later still (circa 1840, according to Merriam-Webster).

This chronology suggests that physic and physician were well established from their Latin root of "natural science" by the time physics and physicist showed up to share dictionary space with them. So perhaps the more searching question would be, How did a root term that meant "nature" and that had long been associated with the natural science of medicine gain a separate meaning that focused on (as MW puts it) "matter, energy, and their interactions"?

With regard to that question, I can only suppose that eighteenth-century European scientists considered that the Greek root word for nature was strong enough to accommodate a second, separate branch (to confuse the imagery a bit) of scientific inquiry.

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UPDATE: Two supplemental points from commenters

Two commenters have provided valuable additional information in response to my original answer. Because comments aren't preserved in the same way that answers are, I want to make sure that these don't get lost. First, in response to my speculation about how European scientists came to adapt the Latin physica and the Greek physikos, Frank points out that the OED reports a remarkable 1840 discussion of physician versus physicist:

1840 Whewell Philos. Induct. Sci. Pref. 71 We might perhaps still use physician as the equivalent of the French physicien‥but probably it would be better to coin a new word. Thus we may say that‥the Physicist proceeds upon the ideas of force, matter, and the properties of matter.

But prior to that physicist had been used to mean “One versed in medical science”:

1716 M. Davies Athen. Brit. III. Diss. Physick 12 Anatomists, Naturalists, Physicists, Medicinist

Second, regarding the same issue of the shared Greek root of physic and physics, Janus Bahs Jacquet offers these thoughts:

[I]t is noteworthy that the Greek root is (in Greek) transparently connected with the base noun φύσις and its underlying verb φύειν, both of which refer to making something grow, creating something, bringing something into being. The underlying root is PIE *bheuH-, which simply means ‘be’, so the ultimate sense of physic(al) is just ‘relating to that which is’. Not a stretch to go from there to ‘nature, the natural world’, but also not a stretch to go to ‘matter, essence, energy’.

My thanks to both of these commenters for their insights.