Where there doctors in the days of Moses following and treating sick people? The Torah tell us that Hashem did heal people.But where there some doctors among them ?
Try this book:
Fred Rosner, Medicine in the Bible and the Talmud: Selection from Classical Jewish Sources. Augmented Edition, LJLE 5 (Hoboken NJ: Ktav Publishing House Inc./Yeshiva University Press, 1995).
In parshas Mishpatim chapter 21 verse 19 we find the term ורפא ירפא, someone who causes someone else damage must pay for his treatment to be healed.
This is a clear reference to human healing.
In fact the Talmud in Bava Kamma 85a uses this to prove that a doctor has permission to heal. Torah Temimah there explains that these words teach us that we should not think we are going against the will of God by healing someone He made sick.
Moses, himself, was a doctor! See B'midbar 21:9 and the verses preceding it. Fiery snakes bit much of the nation, and many were dying. Moses fashions a copper snake, and when people looked at it, they lived.
Granted, that Moses followed G-d's advice. The point is, that one can view this as being no different than a current doctor asking another doctor for advice on how to heal people. Even with the Talmud's interpretation of how this worked (I think it's in Talmud Megillah), the point is that in this incident, Moses still healed people. At least for this incident, it makes him a doctor.