What is the ruling of Islamic law regarding a young man continuing to converse with a girl he proposed to and whose family refused him, and she is now engaged to another? And what is his role in severing this relationship?

Your conversation and relationship with that girl were forbidden to you. If you continue speaking with her, you will be sinful and following the footsteps of Satan, for unlawful relationships begin with conversation. You must fear Allah and repent to Him. If she calls you again, do not answer her.