“If your brother sins against you, go and show him his fault, just between the two of you. If he listens to you, you have won your brother over. But if he will not listen, take one or two others along so that every matter may be established by the testimony of two or three witnesses” (Matthew 18:15-19). This is the biblical approach to resolving conflicts.
First, approach your friend confidentially to provide an opportunity for your friend to admit to wrongdoing and to offer an apology. If that happens, then the encounter was truly a success. If not, then it is time to get some spiritually mature partners to accompany you in approaching your friend for a second time. Be sure you take each step in order. It is unbiblical to recruit the help of outsiders before you have tried to reconcile on your own.
Remember, as long as you do what is right in the sight of God and handle yourself appropriately with honor then you are blameless in the conflict. However, if you allow a root of bitterness to take hold, fault will be found in yourself as well as your friend. The goal should always be restoration and healing approached with sincerity and love. If your friend behaves self-righteously and proudly and will not admit to his or her fault then at least you have tried to work things out according to biblical guidelines. It is your friend who will have to answer to God for being disobedient and self-righteous.