Dear {Name},
I heard that you were upset over {incident}. I'm sorry you feel that way.
I stand by what I said, but I should have chosen different words. I forgot how sensitive you are. Mistakes were made.
While I can't really accept responsibility for what happened, there are others in this {office/household} who felt strongly that I reach out to you on this matter. It's important that we {keep the peace/continue our business arrangement/etc.}. My intent was not to harm you in any way, and it's regrettable that my comments were taken out of context and misconstrued.
Sincerely,
{Name}
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