TL;DR: Almost never.

Men and women treat apologies very differently, both as issuers and recipients.

Men tend to use apologies to try to superficially correct the sudden imbalance of power caused by an insult or impropriety. In other words, if you run into a guy on the street and spill coffee all over his shirt, you've taken away (albeit unintentionally) the power he has over both his personal space and his physical appearance. An apology is an admission of guilt, and thus an abdication of power, designed to correct the inequity created by the incident. The majority of men (men who aren't sociopaths anyway) recognize this for what it is: an earnest attempt to make things right. Most often, an apology from one man to another will be met with a modicum of respect, or at least acknowledgment of intent.

Women tend to use apologies in a decidedly different manner. It is a popular notion within TRP that women do not have as objective and internalized a sense of morality as men do. In other words, right and wrong for women is based less on an objective appraisal of the costs and benefits of actions to one party or another, and more upon whatever her current emotional state is at the time of events transpiring. When a woman is angry, for example, she will tend to be more focused on rationalizing her anger as justifiable, by whatever means necessary, in order to avoid cognitive dissonance, than in discerning whether or not she should actually be angry based on objective facts. "If it makes me feel bad, then it must be bad." When faced with a situation that may merit her issuing an apology, her decision will likely be influenced much more by the chance of losing something of value (social standing, a relationship, material possessions), than by a duty to maintain a stable moral equilibrium.

Men should view a woman on the receiving end of an apology through this same lens. The majority of the time, a woman demanding or requesting an apology is not doing so out of desire for moral rectitude or karmic balance. To a woman on the end of an impropriety, the important thing is not that power has been taken out of her hands, it's not what was actually gained or actually lost; it's that you made her feel bad. You affected her feelings in a negative way, and her feelings are the infallible compass by which she navigates the world; therefore, you must have done something wrong, and her anger is warranted.

This represents a critical fork in the road.

Maybe she is actually in the right, or maybe she is just a little girl throwing a fee-fee tantrum.

While it is important that you ultimately be able to discern one from the other, your response should be more-or-less consistent, no matter what her attitude on the surface indicates. Women are masters of disguising emotion and intent with speech and body language. Maybe she is fuming mad, screaming at you and demanding an apology. Maybe she's calmly stating her case, asserting that you are in the wrong, and she is owed an apology. Maybe she is using the latter to disguise the former. You can't really know; you're placing a bet in the blind.

Whatever the case, when faced with a situation that might merit the issuance of an apology to a woman, you should forego the thought process you apply to a conflict with another man: "have I wronged this man morally, and would an apology help make things right?" She is probably not thinking about that. Regardless of objective right and wrong, you have made her feel bad, so it is your fault, and now she thinks she wants you to fix her feelings. She doesn't.

When faced with a woman demanding an apology because she feels she has been emotionally wronged, asking for a solution she doesn't actually want, you should double down on her female bullshit by playing the game at which you excel. View the situation from a purely objective, cost-benefit perspective. Ask yourself:

  1. What do I stand to gain by not apologizing to her? Let's all collectively admit something. Whether you're totally right or totally wrong, it feels fucking good to stand your ground. We come from an evolutionary background in which, irrespective of morality, we often stood to gain wealth and comfort by kicking the shit out of those who opposed us, and we stood to lose our lives by not doing so. I'm of the opinion that the drive to say "fuck that and fuck you, I'm not moving an inch" is pretty deeply and purposefully ingrained in us and is usually pretty satisfying when we do so. But this is a fleeting, emotional concern that we shouldn't let unduly affect our decision-making process.
  2. What do I stand to lose by not apologizing to her? Think about it. What do you really stand to lose? A week or so worth of sex? If a woman can hold that over your head, it's your fault for not having other options on the back burner and letting her think The Fuck Stops with Her. Refuses to cook you meals? If a woman can hold that over your head, it's your fault for not having acquired that skill yourself. Kicks you out of the house? If you lean on a woman you're seeing for a place of residence, you have fucked up mightily and it is your fault. Any blow-back you might face is not a result of your lack of apology; it's because you failed to cultivate sufficient power and options in the first place.
  3. What do I stand to gain by apologizing to her? Nothing. Literally fucking nothing. An apology to a woman (and really an apology in general) doesn't fix anything. They don't solve problems. They don't un-wreck cars. They don't un-fuck secretaries. They don't redistribute alimony and child support. The don't rebuild broken families. You know what happens right after you say "sorry"? You're sorrier than you were a second ago. That's it.
  4. What do I stand to lose by apologizing to her? This is the big one. The quiet one. The one that no one ever thinks about. What do you stand to lose by saying "I'm sorry", two little words, a total of 3 syllables, to a woman? In a word: Frame.

Apologies between men exist to maintain maintain the balance of power, and thus peace and stability in society. That guy you spilled coffee on? If you smirked and walked away without saying anything, absent the existence of a police and the court system (relatively recent developments), he could pick up a rock and bash your skull in with it. Then your friends and relatives would seek revenge upon him. Then his friends and relatives would seek revenge upon yours. But why go to all that trouble over some coffee and a shirt? An apology between men is an important social convention, a mode of diplomacy that is the first step in preventing petty conflicts between men from reducing stable societies into a dystopian hellscapes of violence and retribution.

But when dealing with a similar situation in a male-female context, remember that that same balance of power does not exist between men and women. Female mental firmware is designed around one consistent premise: that men should always be in charge. Women only seek out and value men who have the balance power firmly in their favor. It is their first and most powerful arousal trigger, and absent that value, her attraction will cease to exist. To a degree determined by the context of the specific interaction, every time you apologize to a woman with whom you are involved, you are acknowledging yourself as her peer rather than her superior, and throwing away the power you have over her that makes you attractive. With regard to men, women view power as a less-than-zero-sum game. When she is able to usurp power from you, not only did you lose every ounce of that power you had, overall she views you as even weaker because she (a frail, indecisive little girl) was able to take it from you. When faced with an apology, most men fall back upon their well-honed and internalized sense of right and wrong. Show a man your belly in order to right a wrong, and he will typically will acknowledge your effort to make things right. Show a woman your belly to do the same, and you can rest assured her first thought, way back in the quiet, limbic recesses of her brain, will be "Wow...I could rip his guts out right now...yuck."

Not that she'll necessarily go through with it, but it will weaken your standing in her mind. Not all women do that, but all women are like that.

I'm not saying never apologize to a woman. The only absolute I will ever put forth is that you should never deal in absolutes.

Just be judicious with your words and behavior.

Whenever you think you may have wronged a woman and owe her an apology, just remember that they play a different game, by different rules, seeking different goals, and that those two little words don't mean the same thing to her as they do to you.