What is the ruling on shunning a friend who thinks I am arrogant, and what are the limits of interaction with him, balancing between the fear of God and self-respect?

In summary: You should initiate the greeting of peace so that you may be the better of the two who have severed ties. You have done well by reminding your friend of the seriousness of estrangement, severance, and abandonment, and you must continue on this path, as Allah the Almighty says: "And not equal are the good deed and the bad. Repel [evil] with that which is better, and thereupon the one whom between you and him is enmity will become as though he was a devoted friend. But none is granted it except those who are patient, and none is granted it except one having a great portion [of good]." Do not let "self-pride" prevent you from initiating, and be the better of the estranged who begins with the greeting of peace. And remember that the gates of Paradise are opened on Mondays and Thursdays, and everyone who does not associate anything with Allah is forgiven, except for one between whom and his brother there is animosity. It is said: "Look at these two until they reconcile." If you initiate the greeting of peace, you will have the reward, and the sin will be upon him if he turns away.