What is the Shar'i ruling on what happened, and what should the questioner do, who committed adultery with a girl who became pregnant by him and gave birth, and the mother desires to marry him to protect her and provide her with cover, but the girl's mother refuses their marriage? So, what is your guidance?

The relationship of love between young men and women is Islamically forbidden. You have committed a grave sin by falling into fornication (zina). The child born from zina is not attributed to you but to her mother. It is permissible for you to marry the girl if both of you sincerely repent. The girl's marriage is conducted through her legal guardian (father). If the closest guardian refuses to marry her to a suitable man, the guardianship transfers to the next in line. If they dispute, the judge marries her. Both of you must repent and increase your good deeds, and try to convince the mother and father. Otherwise, refer the matter to a judge. If that is not possible, then stay away from her and look for someone else.