When you do something wrong forget saying I’m sorry, just do the right thing.
Actions speak louder than words. Doing the right thing lets the other person know that you care enough to change. Will this make everything good? Not necessarily, but we have to ask ourselves one question.
Why is it that this person is upset with you? This individual is upset because they judged your capacity to do the right thing in this particular situation and you CHOSE not to. That very choice can only truly be corrected by choosing to do the right thing in the situations to come.
It is the fear of the habit that you’ll continue to make decisions similar to that initial event that really upsets people. Once you provide evidence that your occasional wrong doing isn’t a habit but merely a lapse in judgement, they have no choice but to forgive you because you allow them to recognize your complete body of work and all of its awesomeness!