At first, I thought “oh come on!”
Then I thought “hey, I should check.”
Now I think “probably not, but it was worth checking”.
medical comes ultimately from Latin mederi “to heal, give medical attention to, cure”: Online Etymology Dictionary. In turn, this ultimately derives from the Indo-European stem *med– (Pokorny’s dictionary), “to measure; to give advice, healing”. The Greek cognate is Homeric μέδομαι “provide for, be mindful of”, and μήδεα “counsels”; the other Latin cognate is meditari “think or reflect on, consider”. The English cognate is to mete out.
Oh, the other Greek cognate? The name suffix –medes. As in Ἀρχιμήδης “Archimedes”.
Looks like Medes, doesn’t it. So where do Medes come from?
Online Etymology Dictionary has the unadventurous suggestion “from king Medos”. Blah, that doesn’t mean anything.
Wikipedia offers: Medes
The original source for different words used to call the Median people, their language and homeland is a directly transmitted Old Iranian geographical name which is attested as the Old Persian “Māda-” (sing. masc.). The meaning of this word is not precisely established. The linguist W. Skalmowski proposes a relation with the proto-Indo European word “med(h)-” meaning “central, suited in the middle” by referring to Old Indic “madhya-” and Old Iranian “maidiia-” both carrying the same meaning and having descendants including Latin medium, Greek méso, and German mittel.
That’s Pokorny’s dictionary : *medhi-. It looks like *med– , but is not the same.
So: medicos are those who mete out healthcare; the Medes are the guys who live in the mid part of Persia. And if that proposal is right, the similarity is coincidental. But for all we know, that proposal might be wrong…