What is the ruling on marrying a girl whom a young man met through social media and with whom he developed a romantic relationship, then the young man repented and regretted it, but he still wishes to marry her because he is unable to forget her?

You did well by ceasing communication with the girl, and by showing remorse and repentance. We hope that Allah accepts it from you.

If the girl is religious and of good character, then hasten to propose to her and marry her, especially given the attachment between you. The Prophet, peace and blessings be upon him, said: "There is nothing like marriage for two who love each other."

Ascertain her situation by asking trustworthy individuals about her. If they praise her highly, then seek Allah's guidance (perform Istikhara).

If marriage to her is facilitated, then all praise is due to Allah. Otherwise, leave her and look for someone else. Ask Allah to facilitate someone better for you, and strive to forget her.