Is it permissible for a Muslim to cut ties with another Muslim who has wronged him and cursed him with the worst insults, and would he be sinful for doing so?

If associating with a person causes harm to one's religion, worldly life, or self, then there is no sin in abandoning them for more than three nights. For sometimes, a beautiful abandonment is better than a harmful association. Nevertheless, it is preferable to investigate the cause of the problem and address it.