Honest question: what’s the “rule” with apologies? Do you owe them even if you strongly feel you’ve done nothing wrong? Like if someone wants one, it often feels like you “better give it or else”.
When I take offence, people can take all the time in the world to explain why I shouldn't be offended, but at the end of the day, I am, and an apology helps. As many HR people have told me, it doesn't matter how I intend to be perceived, what matters is how I *am* perceived.