This is a really interesting question. The meanings are similar, but not quite the same. I think your teacher is correct.
According to and regarding can have similar meanings. One meaning of according to is:
3 if something varies according to a changing factor, it varies in a way that is determined by this factor (Macmillan)
For example:
When getting ready for a long jog, choose your clothes according to the weather.
This means a jogger should wear lightweight clothing in the heat, and warmer clothing in the colder weather.
With that in mind, I suppose you could say:
The doctor decided to operate according to the patient's condition.
However, that's a somewhat odd sentence. I'd interpret it to mean that the way the doctor chose to operate varied somehow, depending on the condition of the patient (for example, some patients require full anesthesia, while others only require a local anesthetic). I think that sentence seems even more strange with the according to is moved to the front of the sentence:
According to the patient's condition, the doctor decided to operate.
Looking at all three of Macmillan's definitions of according to, none of them seem to be a good fit:
1 if you do something according to a plan, system, or set of rules, you do it in a way that agrees with or obeys that plan, system, or set of rules
2 used for saying where information or ideas have come from
Regarding is a more general word that means concerning:
Regarding the patient's condition, the doctor decided to operate.
This means that the doctor evaluated the patient's condition, and decided to operate.