Not civil and Christian. But Civil and personal.
The Jews could not legally put the woman to death as they had no civil jurisdiction.
Genesis 9:6 was written by the One know as the Spirit of Christ.
No contraditction; no dilemma.
As for those stoning the woman, Christ did not distinguish between what was a civil matter and what was sin.
He didn't say, according to the account, under what civil or governmental role was she to be stoned, but she was to be stoned according to the laws brought forward by Moses on behalf of God.
It is then Jesus states, he who is without sin cast the first stone. There in lies the message: not the law and whether it was just or legal, but rather who can condemn the woman to death because they are sinless.
He later states that the Pharisees are judging according to the flesh (to condemn her life).