I am really beating myself up for embarrassing you the other day. I hope you know that it was never my intention to cause you any discomfort.
I never should have {mentioned private issue/made inappropriate joke/etc.} I really stuck my foot in my mouth, and you were the unfortunate victim of my jackassery.
I'm so sorry that I didn't hold my tongue. I hope you will forgive me. I promise that I will behave in such a way that I can be allowed to mingle with polite company from here on out!
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Index of letter of apology templates