What is the ruling on greeting a Christian woman who extends her hand, knowing that the Muslim man does not initiate greetings with them?

It is not permissible to shake hands with a non-mahram woman, whether she is Muslim or non-Muslim, married or unmarried. Whoever has done so must repent and seek forgiveness, and refuse to shake hands in the future. One can gently apologize to the woman, explaining that it is not permissible Islamically.