One reason you don't apologize for someone getting emotional about what you said is that their reaction is not in your control. But here is the skinny from The 16 Commandments of Poon.

VIII. Say you’re sorry only when absolutely necessary

Do not say you’re sorry for every wrong thing you do. It is a posture of submission that no man should reflexively adopt, no matter how alpha he is. Apologizing increases the demand for more apologies. She will come to expect your contrition, like a cat expects its meal at a set time each day. And then your value will lower in her eyes. Instead, if you have done something wrong, you should acknowledge your guilt in a glancing way without resorting to the actual words “I’m sorry.” Pull the Bill Clinton maneuver and say “Mistakes were made” or tell her you “feel bad” about what you did. You are granted two freebie “I’m sorry”s for the life of your relationship; use them wisely.

"I am sorry you are upset/offended". You don't apologize if you are sure you are right. You just display some sympathy for their reaction to the truth/situation, them being upset is their problem and you validate their emotion as a consolation (women crave validation for their emotions/thoughts). This is chipping at your frame, so use sparingly or when absolutely necessary.

However if you realize you are wrong, you apologize and provide a fix/consolation in the same breath to move on from your mistake. You still control the situation, even though you were wrong.

I don't apologize anymore. It's useless. People don't care about apologizes.

Nowadays, I just recognize my mistakes. "You're right I did XXX and shouldn't have. I have no excuses. I'll work it out". That's what a man does. Not whimp into excuses and find an easy way out, just recognize what wrong he made and fix it.

It very few extreme cases, you may apologies.

this woman was giving me shit one day about what i did somewhere and i asked "do you want an apology?" she said "eww god no, yuck!"

I agree with what everyone else is saying here, but there's another side to it that makes it a much more tricky thing.

In the eyes of some people, "I'm sorry" means, "I apologize". In the eyes of other people, it means, "I sympathize with you." You know how Canadians have a stereotype of saying, "I'm sorry" all the time? It's not that they are taking ownership for your problem, it's a way of expressing sympathy with you and being polite. Women often want to hear "I'm sorry" as a comfort test rather than expecting you to admit blame or help solve their problems.

So my advice is what everyone else is saying, only utter apologies when you truly are to blame, and don't use the phrase "I'm sorry" because there are many ways to take it so it's best to be direct and clear. If you're trying to sympathize, you can say something that can't be taken any other way like, "It's too bad your day sucked at work, let's go out for a walk at the park and focus on something else."

If you apologize just for hurting her feelings, even if it was unintentional, she'll start to expect it.

Apologize if you know you well and truly fucked up, but no other reason. There are times when a man should own his mistakes but here isn't one of them. It will bite you I'm the ass in the long run.

After an hour of no input, I'll try to start a conversation. IMO "I apologize" > "I am sorry" Why?

It might be due to an under usage of the term compared to the latter. Unique, possibly interpreted as more intelligent. Possibly.

"I'm sorry" has been said to girls from past betas no doubt. Thus emotionally connecting you to a previous beta?

"I apologize" is a verb, a concise action. While on the other hand "I am sorry" is a an opening to ones emotional state.

Also, the ending sound of apologize, IMO, sounds more masculine with a pretty strong sounding conclusion. More difficult to drag out, possibly eliminating indecisiveness? Which is seen as unattractive to women and men alike.

"I'm sorry" on the other hand, can become a whiny "I am sorryyyy..."

Well anyway. Thoughts? Input?

The simple approach: be true to YOURSELF.

If you did something that you sincerely regret based on your own moral compass, apologize for that and only that.

Otherwise, do not apologize for someone else's emotions. They are responsible for their own emotions. Also, it is poor integrity to apologize for something you don't actually regret. That's called lying.

I feel that if you look at this from an integrity perspective, it becomes very simple.

It's better to under-apologize than to over-apologize. Deep down, if someone respects you, likes you, and holds you in good regard, they won't get so easily offended if you slip up and say something wrong. It would take more than one time.

If someone doesn't like you, doesn't respect you, it doesn't matter what you say or do, they'll FIND something to be offended about. In that case, it's even more important to maintain your frame.

Never say "Sorry,"

Simply use "mistakes were made."

That is a poon commandment to never assume the submissive role

Nope.

You can't go round basing your actions on other people's feelings esp. W/o having I tent to do the harm perceived in the first place. That aint RP, your mama will tell you that.

Think about why they would be offended. Is what you said truly stupid and hurtful to others? Or is the hamster running full steam and you bruised her ego in an honest, playful way?  

The best way to look at it is like this: if you made an honest mistake, be a man and apologize. Otherwise, stand up for yourself like a man would.