Is what I did—chatting with a man online, developing a love relationship with him, then taking his number and contacting him—a sin that warrants blame, and what should I do to convince my family of him after their rejection due to his nationality and their objection to online relationships?

You have committed Islamic transgressions by speaking with a non-mahram man and exchanging romantic messages and phone conversations with him. There are no love relationships outside of marriage in Islam. Your family is right in rejecting this illegitimate relationship.

If the man desires marriage, he should propose directly. We advise you to repent to Allah and distance yourself from this man, and he too should repent. If he wishes to marry, then let him approach houses through their proper doors.