.
"Instead, the reason we apologize is to make the person we intentionally or unintentionally harmed feel better, not to make ourselves feel better."
Until it's forced at least. Once it's forced the nature of it changes entirely.
"An apology means "I see you were harmed by my action, and that matters to me""
It tends to mean that just as often as "I see I've worked myself into a jam here, and this seems to be the easiest solution". You know what would make that become less of a possibility? Oppressive people not forcing others to go through this act when it's not what they would have done on their own. Suggesting at least still gives them the power of choice.
"And how do you know in advance that the person is not going to perceive it as cruel action? or how do you know if after the person feel it as cruel offense but never tell you?""
If it matters to me, their character will usually betray that when I'm gauging for how to talk to the person. If it doesn't matter, then I don't care. I figure if it's someone who is going to be around me then they better get used to who I am. It sorts itself out like a natural filter.
I'm not going to live my life according to others' expectations unless they are somehow my better, and others should be granted that same liberty.
"or even if the person tells you and ask an apology?"
Someone asking for an apology, more often than not, doesn't deserve it. If I didn't apologize of my own volition, then I'm not planning on it just because of someone else's insistence. At best it'd only make sense if something I did didn't occur to me as being a "bad thing" to do, but usually I'll see how it might affect others in advance.
Needing an apology over little things usually just betrays how sensitive they are.
"according to you this kind of apology isn't valid because someone else is asking it to you and provoking it."
Bingo. It should be their own idea, not insisted from others. Good manners should be a privilege, not an expectation.
"But it was a polite remark."
Didn't read like one. It read like a demand from someone not used to making demands.
"It basically was a comment with my opinion, not an imposition or do you see it in a different way and why?"
You were kinda telling him what to do. Comments aren't commands.
"So I can't have an opinion? or is because is my opinion it has to be wrong?"
You can, but when your opinion becomes how you view others should be to the point of demands? It risks becoming oppressive. An opinion isn't "Apologize to her", an opinion is "I'd apologize to her if I were you."
"Or maybe are you trying to said it was a private discussion in a public post. My bad then."
Mostly saying, is it your business how those two interact with each other? Is it your business if he apologizes? Have you ever thought she might not need or maybe even want to be defended? What made you feel the need to step in, something as sexist as chivalry? Who are you to tell someone who to be and how to act, and who are you to judge when someone is too weak to defend themselves? Let them be rude if they want to be, just how I'll let you be rude beyond what I figure on your part to be an "error" if you happen to continue this pattern.
"Did you saw anyone complaining about everything?"
Her quote imposes an idea of shame upon not speaking up when you feel the urge. Complaining every time that nagging itch needs scratching will only serve to divide your word's worth into all of those statements instead of jam packing it into the times where they really ought to matter. This "injustice" was nothing.
"So this is an opinion according to you??"
What isn't an opinion?
"And again why do you think I was forcing him."
You begin by berating him, which is fine, that's when it was still your opinion backed by some research. Then came "Then when you undestand what you did, back and ask apologies to Stayonhere." Unless I misread it, it both shows that you see him as beneath you and that you feel the authority to tell him what to do just because he did what you view as "wrong". Politeness isn't mandatory.
"This is about life and human history. Maybe a discussion for another time."
Actually it's about you.
""I'm not attacking you. (if you are counting me too)I just told you just what I think. (nothing more nothing less.)"
Just because you aim to use that to cover your ass doesn't change what it was, but hey, you apologized, so I guess it's "all better" right? 
"People can figure some things by themselves. But people just cannot figure some other things by themselves. They need from others to guide or to provide what they lack."
Who made that your job? There's no one telling you to make people become "more civilized". When you separate yourself from other people's ignorance and misunderstandings is when you make yourself out to be better than they are. You have no authority over what they know beyond granting your own perspective, for your perspective is likely just as off base as theirs for different reasons. Your very means of "educating" people could be spreading poison you don't even recognize it being.
"Do apologies serve a useful purpose? Or are they just power plays between people?"
They're power plays if they're forced, they're admissions of guilt if brought on unprovoked.
"And that gives you the right to be cruel or unkind and treat others as you want, ignoring the rules of courtesy and humility?"
It's only cruel if they take it as such, for the cruelty of it is at best only premeditated through intent. If the person hearing it doesn't take it as an offense, is it even cruel anymore? No, cruelty is based purely on how it's perceived, not the actions themselves. If someone cares to be nice, they ought to be doing so because it's what they want to do, not because it's the "right thing to do".
“Courtesy is a silver lining around the dark clouds of civilization; it is the best part of refinement and in many ways, an art of heroic beauty in the vast gallery of man's cruelty and baseness.â€
And yet you aren't showing the courtesy of allowing him to be himself, insisting he be something else simply because you view him as rude. You telling him who he should or shouldn't be is no courtesy, it's an imposition, it's you telling him that you think he needs to be a specific person, and to be otherwise is "wrong". You're choosing to judge him in a fashion that was destined to pick a fight instead of simply choosing to question his motives in the name of understanding or converting him slowly into your way of thinking.
In essence, by telling him who to be, you have descended to the same level as him, since you're judging him in that fashion for having judged someone else in a similar one.
"but sometimes is necessary and I'm sorry for that."
That is purely your opinion, not fact. By insisting in this fashion you are forcing your views upon him, and that is just as rude as his original view he gave her. You're only rationalizing it simply because he presented himself as a "villain" of sorts by daring to be rude first, but his having "done it first" doesn't change the nature of your response to him.
"I'll tell you one thing I think is worse, be aware of an injustice and do nothing about it."
Is that really an injustice? Really? Come on, don't be so weak. At the very least let her defend herself instead of taking the responsibility upon yourself for reasons beyond petty enjoyment.
""There may be times when we are powerless to prevent injustice, but there must never be a time when we fail to protest."Elie Wiesel."
Bullshit, there's a time and a place for everything. If you complain about everything, then your complaints will become expected. When anything is "expected" it carries less weight from having it become more about you than the issue itself. It's wiser to choose when to speak up instead of speaking up in every single case, otherwise you just make yourself out to be whiny.
"and yes I'm agreed is should not be an obligation but a duty of any person who lives and wants to live in a civilized society or interact with another human being in what must be a civilized way."
And you think trying to force someone to apologize will accomplish this? How is presenting yourself in a brutish, forceful fashion, supposed to inspire them to not do that? You're fighting fire with fire with a misplaced sense of justice as your justification to do exactly what he was doing in different words. If you want your "opinion" to be respected, you must first bother to respect and play along with his, otherwise you're just building a wall between the two of you.
"Just have to be aware you did wrong and we willingly to take reponsability and try to fix your mistakes."
If they don't see what they did as wrong, why should they feel the need to apologize? Your views are not the views every human being will follow, so what business of yours is it to impose it upon others? I understand trying to show him why he ought to, but to insist forcefully that he ought to do it himself? At it's core it's remarkably rude, and completely denies him the right to his own perspective.
"But sometimes you should take the risk doing it anyway, to do what can become in a greater good, risking to become the villain."
Greater good is a myth, a common excuse for atrocities. "The road to hell is paved with good intentions."
Do you seriously not see how your own views are extremely forceful of views they may not share? To force things upon people as if it's the only way to look at something is to allow yourself to be a bigot. Is that really what you want to be? If yes, this discussion is done now, because there will be no means of reasoning with you.
"1-. About apologizing: Yes indeed people think to apologize is a weakness."
It's a form of strength to apologize on your own, for apologizing reveals vulnerabilities, but to be forced into an apology instead has them just being the one forcing it's bitch. Hint at it, request it, but pushing it like it's the only answer only reveals how limited your own world view is.
"In other word apologizing and try to fix the damage when you have to, show you are reasonable enough and have the courage to take responsibility for your actions."
Only when it's their own idea. Think of the teacher that forces a child to apologize for something they felt fully justified doing. Are they learning anything other than how to con an authority figure? Probability leans towards "No" unless they have some reason to respect that authority figure beyond their role.
"2-. An imposition: Well I still feel it was not an imposition but a well-intentioned remark."
You were outright telling him what to do. Reread your words and try to tell me what you were insisting upon wasn't forceful, I dare you.
"although some people may find their own ways, others need guidance in the form of a well-intentioned remark or even an imposition as well "the end justifies the means"."
People need to figure things out for themselves, make their own mistakes. If their experience is anything but that it's only half informed. Sure you can give them your two cents, just like anyone else, but to insist they change their own lives just because you don't like it is no better than anyone else doing the same thing to you.
"Yes, I think you are right."
Then why is that what you're going with here? Why not a different approach?