What is the Islamic legal ruling on the relationship between the questioner and the son who bears his name, and is what he did—marrying the child's mother after committing zina with her, and then divorcing her—Islamically permissible?

Your marriage to the Christian woman is invalid because she is not chaste and practices prostitution. The Qur’an stipulated chastity as a condition for the permissibility of marrying a Christian woman. As for the child, he is not legally your son; he is a child of fornication, and the water of fornication is wasted, establishing neither lineage nor inheritance. This is the opinion of the majority of scholars. A child born of fornication is not attributed to the fornicator, even if he claims him, and you are not obligated to financially support him, especially given your doubt regarding his lineage to you.