If everyone had this mentality, no one would ever talk. Ever.
If you care about someone and you’re willing to swallow your pride to make things work, talk to them. I remember one of my friends really hurt me and I gave him the cold shoulder. He could’ve easily returned my coldness, insisted he was right and never tried to repair our friendship, but instead he came up to me and apologized, and we’re still close friends today.
It takes one person to break the cycle, but if neither person is willing to step off their high horse, you could lose a friendship forever.