What is the ruling on a relationship that developed online between a young man and a young woman who only communicated in writing, and is it permissible for them to resume communicating in writing with the intention of getting married?

Among the objectives of Sharia is to block the means leading to the forbidden (haram). Speaking with and messaging a young woman may lead to falling into forbidden acts. Jurists have prohibited speaking with a young non-mahram woman without necessity, out of fear of temptation. For example, the Hanafis believe it is impermissible, and the Hanbalis hold that a young woman should not return a greeting of peace (salam) to a man to avert corruption. As for the Shafi'is, they deem it forbidden for a young woman to initiate a greeting of peace to a man or to return his greeting, and they consider it disliked for a man to initiate a greeting to her. Therefore, it is impermissible for the brother to continue conversing with the girl online, and he must repent. If he desires her as a wife, he should strive to arrange the matters of marriage. As for the verse in Surah Al-Baqarah, it pertains to the indirect proposal (ta'rid) to a woman in her 'iddah after her husband's death, not a direct proposal (tasrih). There is a significant difference between a man speaking with a woman for the purpose of proposing marriage and speaking beyond that; the former is permissible out of necessity.