“I’m sorry you feel that way” isn’t an apology. It’s a refund on your responsibility.

That sentence moves the subject from what you did to how they reacted, which leaves you with nothing to answer for. It sounds polite, which is why it stings: the shape of an apology with the content taken out. What they were waiting for was a line with you as the subject.

If you reach for it, stop and say the simpler thing, which is what you did, without a word about how they took it. You can still think they overreacted and apologize for your part at the same time. If you truly believe you did nothing, say that plainly instead of dressing it as regret.