I work a lot with couples, and there are often hurt feelings and pain between them, where both feel like they've been wronged, and they are each waiting for the other to apologize. When they are asked to apologize, they will often say that they didn't do anything wrong and shouldn't have to apologize.
I often advise people to not wait for the other person to change, but rather to change themselves first. In other words, control what you can control (yourself), rather than another person, who is out of your control.
If you're willing to take the first step to resolve a conflict between you and another person, here's what I'd suggest.
First, make sure you understand where the other person is coming from. Ask this person to explain the pain to you. Ask clarifying questions if necessary. Try to restate your understanding of their unhappiness, without agreeing or disagreeing to the premise, and ask if you got it right.
You might say "You are angry with me because didn't feel like I listened to your concerns, and you think I just pushed my own agenda without taking your opinions seriously," or "You're really lonely and hurt because I don't seem interested in physical intimacy with you." Then follow your summary with "Did I get that right? Is there anything you'd want to correct or add?"
Do not use this time to tell them why they are not accurately perceiving the situation, defend yourself, tell your side, or do anything besides just clarify your comprehension.
Once you have fully understood their point of view, you can try to offer compassion. Again, you are not agreeing with the accuracy of their perceptions, you are only imagining how they might feel if that's how they experienced the situation. You can probably imagine how painful it can be to feel hurt, sad, angry, lonely, or whatever else the other person is feeling.
Now, once you understand their experience, and have some compassion for their pain, you can apologize. To be clear, you are not accepting blame, and you are not agreeing with their version of reality. You are being regretful that someone you care about is suffering as a result of your interactions together.
Here are a few examples of what you might say: "You really felt judged by me yesterday. I'm so sorry you felt so hurt," or "I am so sorry that our fight left you feeling so misunderstood," or "It was never my intention to cause you pain, and I am really regretful that our interactions hurt you so deeply."
Note that this is not "I'm sorry you feel that way, but..." or "sorry, not sorry." It's also not just giving in and apologizing to make the problem go away. Both of these approaches rarely bring healing or intimacy, but rather increase the distance and resentment.
I invite you to experiment with this approach in a disagreement you're having with someone who is close to you. See if you can listen openly, imagine their suffering, and offer sorrow, regret and/or compassion for the pain this person is having. Give it a try, and let me know how it goes.