why is the doctor civil to Silver and calmly committed to treating the wounded?

Because it's his moral duty...........

Early the next morning, Dr. Livesey appears at the stockade in order to take care of those men that are his patients. Long John Silver informs the doctor of Jim's presence and the doctor just nods grimly to Jim and proceeds to his patients, treating them professionally and showing absolutely no fear about being in the midst of the enemy.

The major thing to remember is that his good fortune is due as much to luck as to skill. That is evident, even to the other characters, in this chapter as Dr. Livesey tells him, "There is a kind of fate in this. Every step, it's you that saves our life."

This was a part of the bargain they made when the Squire and his

friends left the stockade. Furthermore, the doctor says that it is a point of honor to save them so that they receive their just punishment under the law (the gallows).