I apologize all the time.
I tend to "shoot from the hip". The upside of that is that you never have to wonder where I stand or if I am being honest with you. The downside is that I will sometimes spout of at the mouth (or keyboard) before completely thinking something through. When called out on it, I will usually immediately admit I was wrong and apologize. Or at least I will admit that my tone was inappropriate even if I stand by my opinion.
There is no need, in my view, to "double down" to save your pride. Usually, the tension will go away more quickly if you just "fess up" and truly apologize.
Yesterday in church my pastor did a great lesson on being a peacemaker in your house, and in your circle of influence. It's not about being a doormat. It is, however, about taking stock of where you are in a conversation, and more broadly in a relationship, and weighing your options as to what could be gained (or destroyed) by forcing any one issue. (That's as far as I am going with this so don't freak out.) It is something I truly need to work on in my own life.
My writing style is very sarcastic and "brash". I have probably apologized to a hundred people here at TB alone for such blatherings. I come across as much more harsh that I feel. There is no shame (to me) in apologizing for stating something in a way that crosses a line for someone else. And that's not a "the problem is you, not me" cop out. I honestly understand that not everyone trash talks like I do. So, they are perfectly within their rights to be put off by something I have said. And I really can perpetuate a good conversation by simply walking back a step, taking a breath, and apologizing for "getting in their face" in a manner of speaking.
@Downunderwonder , @Immigrant and several others can attest to the fact that I will knee jerk on occasion. But I can also sincerely apologize for knee jerking and, in a manner of speaking, beg them to continue a conversation. I have learned a lot from both, and others. They are two guys with whom I disagree on some issues. But they are also thoughtful and interesting people with whom I would love to break bread if the opportunity presented itself.
Edit to add
@GregC as well.
Be who you are. Allow others the same. Apologize when you screw up.
Great topic question.