What is the ruling of the Sharia regarding a young man and woman pledging not to marry except each other, and in the event that the marriage is not completed, that each of them will retain their love for the other?

It is not permissible for a Muslim to establish an unlawful relationship with a woman who is not lawful for him, nor to promise to continue such a relationship, for relations between the sexes in Islam are only through marriage. If love occurs without seeking it, there is no sin in it so long as it does not lead to something forbidden. Promising not to marry anyone other than each other is not appropriate; it goes against the Sunnah of the Prophet, peace and blessings be upon him, for marriage may become obligatory when there is fear of temptation. It is preferable for the girl's family to marry her to someone whose religiosity and character they approve of, if she is inclined towards him. If they insist on refusal, then you should turn your attention away from her and seek someone else for the sake of chastity.