Most likely yes
There's been quite a lot of discussion about this subject in various forums. you can find some here and here. Since nothing on the Core book, nor on any other sourcebooks I'm aware of (the only mention in street grimoire of medicine is in unrelated fluff text), explicitly states which is the case, I'll try to explain why it seems the most logical ruling is that it helps:
- Motivation: The "logical" motivation with the "no medkits or magic" rule is to prevent mages from rapidly recovering from drain while in a run. Medicine does help speed up the healing, but only by making you recover more damage after an hour of rest, not making you recover damage almost instantly, so being able to use it in the middle of a run seems unliekly.
- Consistency: Magical healing and medkits are pitted together almost always, as they both act besides natural healing and extremely fast. Medicine is pretty much always treated like its own thing. Since both are mentioned and medicine is not, it seems that medicine would work.
- Medicine improves, not replaces, natural healing: While the two forms of healing mentioned are independent from natural healing, medicine just makes it so when the natural healing kicks in, it is more effective.
- Medicine healing drain makes sense: While a medkit would be difficult to use to treat headaches/bloody nose from drain fast, and casting magic to heal the damage from casting magic (besides being broken) would be quite counterintuitive, treating it with rest, tea, incense or light drugs seems natural. You are not "healing it", just helping your body recover faster.
So, while there isn't (as far as I know) a definitve, official ruling, letting medicine help heal drain makes the most sense.
Also, many people mentioned the existence of the "Magical Health" specialty in medicine as proof that medicine can help drain. To me it seems this exists to negate the -2 to employ medicine on the awakened, but it could be further proof.