Speaking only from my personal experience...
Most humans (including myself) are stubborn and view differences of opinion as right vs wrong. Nobody enjoys being wrong. What we neglect to understand - especially in the heat of the moment - is that each person comes to the table with different viewpoints, opinions, life experiences, etc...
If you truly want to salvage the relationship, you'll understand that being right or wrong is of very little consequence and what is important is the harmony of the relationship.
Even if you think you're correct, you'll be the one to apologize. You'll be tempted to sneak "ifs" and "buts" in your apology, but don't. Simply apologize for the misunderstanding, express that (not how) your point was different, and explain that you will try better in the future to understand what upsets them because you want to see them happy.
At first it might seem like overkill- do it anyways. Introducing this type of harmony and understanding into the relationship has the unique ability to force the other side to look inwardly and eventually reciprocate. It may take a lot of time, and a lot of apologizing even when you'd normally point your finger to the other person, but be persistent and vigilante in your attempt to self-regulate.
I think this plant the seed that resets that boundaries and terms of the relationship. With so much baggage, people get their roots deeply entrenched in feelings, conversations, moments, etc... and those enter into the picture of new disagreements and understandings even when they should not.
Rip the roots out of the ground. Try to start fresh. You mustn't tell the other person you're doing this- just do it yourself. See if it works.
At the same time this doesn't mean you need to be silent regarding your own displeasures. Just don't express them immediately. Stop yourself before reacting, take a breather, return to the topic when you're clear headed, and calmly say, "I want to talk to you about something important when you have a moment."
Saying something is important to you, even if it's small in the grand scheme of things, may help the other person to be more understanding instead of confrontational. Your goal isn't to get an apology or prove the other person wrong, just to help them understand how something made you feel. If they truly love you, they'll try on their own to learn these things and avoid upsetting you in the future.
All the above sounds pretty straight forward, but the truth is it's hard to do. Because nobody wants to be wrong. Nobody wants to feel vulnerable. But in my opinion, having the ability to admit when you're wrong, show vulnerability, and put the happiness of your significant other and health of your relationship above your own pride is one of the best expressions of love you can show.
This may or may not ring true in your particular relationship, but it's stuff I've taught myself after the course of many, many failed relationships! It's serving me well right now, but everyone is a constant work-in-progress.
On the flip side, it's okay to reach a point in the relationship where you realize it just won't work. A relationship not working often isn't one or the other person's failures, just a difference in compatibility. You can still love that person. And when you do, sometimes letting that person go even though you love them is what's best for both of you.
