To not apologize takes practice, like a musical instrument, or language. You must constantly remove 'sorry' from your speech, just as you should remove, 'uhm,' 'like,' and, 'you know'.
I hope people read this and follow my advice. I used to use sorry as almost a physical reaction, but now it is gone, thankfully.
Instead of apologizing, you must change your inner core, and if you force a mistake on someone, you must reverse it. Actions, not words.