Is the young man's saying to the girl, "Marry me," and her saying, "I accept to marry you," considered a valid and permissible Islamic marriage contract, and what is its ruling?

Marriage has pillars without which it cannot be contracted: the woman's guardian or his representative, two just witnesses, and a formula indicating marriage. The phrase "Marry me" with the woman's acceptance does not constitute a valid marriage, and cohabitation based on it is considered fornication (zina). What is obligatory is to sever ties with the girl and repent for the forbidden acts that occurred. If you wish to marry her, then contract a valid marriage with her.