I disagree with some of the things he says, namely:
‘everything dies; everything has its time’.
This attitude is unlike that of the Doctor in the classic series, who frequently risked his life to help vulnerable enemies. It is best explained as a reflection of the belief that death has to be accepted and even welcomed as part of the process that makes life possible and desirable. In ‘New Earth’ Cassandra comes to accept her own inevitable death.
There are some problems with this:
1) Cassandra was a SHEET OF HUMAN SKIN. Her body had completely fallen apart and was inhabiting a STOLEN BODY. Why would The Doctor let her live when she had basically become a parasite?
2) She was a recurring villain.
3) He DOES help her fulfill one last wish. There really wasn't anything else The Doctor could do, short of letting her inhabit someone else's body. He let her have her last wish, thus making her demise as comfortable as possible.