Dear {Name},
I am so sorry for insulting you the other day. I'm truly mortified. I certainly didn't intend to hurt your feelings, but clearly that was the result of my careless words. I see now that my comment was artless and uncalled for. What I should have said was {other thing}.
I hope you will forgive my rudeness. Please know that I hold you in the highest regard and greatly value our {business/personal/family} relationship.
Sincerely,
{Name}
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