Regret and remorse are the results of not thinking before speaking, or speaking when one shouldn't. Don't do either, the default position should be silence, and speech, when needed, should be laconic. I used to tell my kids "Say what you need to say, only what you need to say, and say it in as few words as possible."
That way you avoid unintentional damage. If the damage was inentional, then don't apologise.
We talk too much, all of us.