Must a married Muslim man with children ignore a relationship that resulted in pregnancy with another woman, or ask her to abort the fetus, or marry her?

We have previously discussed the rulings regarding a man who impregnates a woman out of wedlock and the rulings on abortion in Fatwa no. 32758.

If the Greek woman your brother impregnated out of wedlock is from the People of the Book, then marriage to her is permissible, as Allah Almighty says: {and virtuous women from among those who were given the Scripture} [Quran 5:5]. However, if she is a polytheist who does not adhere to a divine scripture, then marriage to her is not permissible, as Allah Almighty says: {And do not marry polytheistic women until they believe} [Quran 2:221].

Your brother is not permitted to leave his two children with their Christian mother, as she will raise them in disbelief. Scholars have preferred the opinion that a mother's right to custody is forfeited if she is a disbeliever, based on Allah Almighty's saying: {And never will Allah grant the disbelievers a way against the believers} [Quran 4:141], and based on the Hadith: "Every child is born upon Fitra (natural inclination to Islam), but his parents make him a Jew, a Christian, or a Magian."

Your brother is advised to return to his homeland and leave the lands of disbelief, for fear of being tempted away from Islam.