Michelle Malkin commends Robert Novak for genuinely apologizing, and points to some other public figures who made "fake apologies": link.
Here's how to apologize:
What I did (specify it) was wrong, and I apologize for doing it.
Here's how to not apologize:
I apologize if what I did hurt your feelings.
Here's why this is a fake apology:
This is not about my action - it's about your reaction.
The fake apologizer is actually lamenting the other party's over-sensitivity, as in:
I regret that you are over-sensitive. If you would have displayed normal sensitivity, this would not have been a problem.
Thus the fake apologizer can continue to believe in the rightness of their action. They can even revel in their cleverness - as the other party often fails to notice their subterfuge.
What if one genuinely regrets that the other party was placed in a predicament, yet genuinely believes their own actions were righteous and correct? Then there is nothing to apologize for. This would be an honest communication - though its a bit harsh:
I correctly chose the best course of action. I empathize with your predicament - but that's the breaks. I would do the same thing again.
An action is either optimal, or suboptimal. A true apology always acknowledges suboptimal action, as in:
I could've created the desired result in a way which was more sensitive to your predicament, and I apologize for not doing so.
Why is apologizing so difficult?
Most people are heavily invested in being right. Their self-image is of being right about things, and of being on the right side of issues. They sometimes associate this with being virtuous. People will choose to be right at the expense of their relationship with a parent, child, or sibling. People will choose to be right at the cost of their jobs. People will choose to be right at the expense of make-up sex with their spouses.
Instead of self-identifying with being right, make a subtle shift in perspective, and self-identify as someone who is committed to certain principles, to certain values, and to achieving certain results. When conflict occurs, declare yourself - as in:
This is what I'm committed to, and this is who I am going to be in this conversation: a person who is committed to x,y,z.
This has the effect of bumping a conversation onto a higher plane, where everyone's true motivations can be seen and acknowledged. For example:
I am committed to our having a close, loving, and nurturing relationship
creates a different conversational dynamic than:
I am committed to you either fixing that leaky pipe, or dying a painful death.
Hopefully, both parties can find common ground on a higher plane, and consider lower plane issues against that backdrop of shared values and goals.
This is not a clever tactic designed to win an argument. In my own life, instances of self-declaration have opened my eyes to who I truly intended to be, but was not being:
I am not a person who is committed to being right. I am a person who is committed to us having a wonderful relationship.
I am committed to our department accomplishing this particular goal, and that's who I'm going to be in this conversation: a person who is committed to our department accomplishing this particular goal.