Somebody wants an apology from you, and here's the complication nobody prepares you for: you don't feel sorry. Not really. Maybe you think you were right. Maybe the thing they're hurt about was a fair, necessary decision that happened to land badly. Maybe you feel bad that they're upset and not at all bad about what you did. So you're stuck between two bad options, the fake apology that lies for the sake of peace, or the refusal that lets the whole thing fester.
Here's the reframe that gets you out: an apology doesn't require you to feel sorry about the decision. It requires you to be honest about the effect. Those are two different things, and conflating them is what's making this feel impossible. "I'm sorry I hurt you" and "I'm sorry I was wrong" are not the same sentence, and only one of them requires you to abandon your position.
Say what happened, not what you think about it. "I know that decision landed hard for you" is true regardless of whether you'd make the same call again. It doesn't concede the point. It just acknowledges the impact, which is often the entire thing the other person needed named.
Separate the outcome from the intention if that split is honest. "I wasn't trying to make things harder for you, and I can see that's what happened" holds both truths at once: your intention was fine, the result wasn't, and you're not going to pretend either one away.
Skip "I'm sorry, but." The word "but" erases everything before it, and everyone knows it. If you have a real defense, it can exist somewhere else, in a separate conversation, not bolted onto the apology where it cancels the acknowledgment out.
Something like:
I know Tuesday's call didn't go the way you wanted, and I can see it's been sitting with you since. That wasn't my intention, and I'm sorry it landed that way. I still think the decision was the right one given what we knew, and I'm glad to talk through that separately if it would help. But I wanted you to know I heard how hard it hit.
That's not a surrender. It's an honest accounting: here's what happened, here's what I intended, here's what actually landed, and I'm not going to pretend the first two make the third one disappear.
What you don't get to do is refuse any acknowledgment at all because you don't feel sorry about the substance. "I did nothing wrong" as your entire response, when someone is genuinely hurt, isn't honesty, it's a refusal to see them. Somewhere in almost every conflict there's a true thing you can say without lying about your convictions. Find that thing and say it. The relationship usually isn't asking you to change your mind. It's asking to be seen.