All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
That young man is obliged to sincerely repent to Allaah because it is not permissible for a Muslim man to have a forbidden relationship with a non-Mahram woman as we clarified in Fatwa 81356.
Indeed, he did well by cutting the relationship with that woman and it is not permissible for him to meet her or talk to her even with the intention of calling her to Islam as this may lead him to commit a sin with her again. That is because he is not religiously obliged to call her to Islam; he does not have any obligation towards her as he believes. Also, the truthfulness of his repentance and its acceptance do not depend on him calling her to Islam. For more benefit on the conditions of repentance, please refer to Fataawa 86527 and 87903.
Moreover, he is obliged to stay away from her as we mentioned whether or not her parents ask him to do so.
As for calling that girl or any other girls to Islam, it could be done by appointing some girls who have religious knowledge to call her or them to Islam. For more benefit, please refer to Fatwa 105117.
Allaah Knows best.