Apologies come in three steps.
Confession: a simple statement of what was done. Call the lie a lie. Call self righteousness what is. Call out the offense such that the offended might see that we too see exactly what has transpired.
Repentance: this is literal heart-break, on account of the pain that we’ve caused, and a consequent turning away from that kind of action.
Forgiveness: this is the one that has to come from the offended. Sometimes offended parties aren’t ready to forgive. Sometimes they’ve already forgiven. Sometimes the confession and display of repentance is the thing that was needed to shift their heart towards forgiveness. Regardless we can’t know for sure before approaching the conversation.
So many worthless things stand in the way of this work. Our reputation. Our fear of how others will respond. Are either of these worth more than a possible reconciled relationship?