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You can't control anybody else but you or their mind. You can make a gesture but in the end it is up to them to accept it. Sometimes things are better left unsaid. You may want to write the letter just saying you are sorry but not go into the details of the event. He has a choice whether to accept your apology or not. The same is true of a verbal apology. If this is heartfelt and you meet him where he is in finding words that express your genuine remorse he has a chance whether to accept that or not. You also have to put a period. You can't feel guilty forever to the extent that it effects your life in a negative way. If you are genuine then he cannot expect more than this. Decide if your apology will be verbal or written and pursue that avenue.