What must be done to keep the child, and will he be lost, and will he bear his name, and is it permissible to care for him, after a foreign woman becomes pregnant with his child through zina and refuses marriage, given his desire to take responsibility?

A Muslim should draw lessons and warnings from stories of falling into pitfalls, for intentions alone are not sufficient to rectify deeds; rather, the deed itself must be sound. As for the matter mentioned in the question:

Firstly: It is not permissible for a Muslim to marry anyone other than People of the Book. Therefore, a girl who does not belong to any religion at all cannot be married, based on the Almighty’s saying: (And do not marry polytheistic women until they believe).

Secondly: A child born out of wedlock (zina) is not attributed to the fornicator and does not inherit from him, as stated in the hadith: (And if it is from a free woman with whom he fornicated, then he is not to be affiliated [to him] nor does he inherit).

Thirdly: It is permissible to spend on a child born out of wedlock if the fornicator desires to do so, provided that this is done remotely (e.g., via bank transfer) to avoid renewing the sin.

Fourthly: It is not permissible to continue trying to convince the girl to marry, nor to use Istikhara (prayer for guidance) or pregnancy as justification for the preference of this marriage, as a child born out of wedlock is not a sign of a lawful marriage. All forms of communication with her must cease. If she is not from the People of the Book, there is no way to marry her, and if she is from the People of the Book, she has already rejected this marriage.