I would choose to write them a letter. That way they have your thoughts of apology in a form that is more longlasting than your spoken words would ever be. They will read your letter once... then read it again later... and then read it again after that... until their heart finally softens and they finally accept your apology.
You can't, it's just that simple.
Don't. Let them cool off before you say anything to them. If they just don't want to plain talk to you at all, don't say anything until they talk to you. If they don't talk to you ever again, sorry won't ever cut it, so you should move on.
Email them a note or write a letter, send a card or something along that line that simply says....I'm sorry, please give me a chance to talk to you and explain. I miss having you in my life.
He might surprise you. At any rate, at least you will know that you tried everything you could and if he's still not willing, then maybe it's time to move on. Sending him a note or something let's him see you are sorry and wanting to talk and will give him a chance to think about it and possibly calm down and maybe then he will be willing to listen and hear your apology. It's worth a try!
Just keep shouting sorry at them.
You say sorry by not talking to them. Because they don't want to talk to you. Learn to let go.
Send a note.