Yes, please believe me, I do understand. They said/did something that hurt you. It made you angry so you just had to hurt them back. Once that pendulum swings, the conflict begins. They hurt you so you hurt them back. Think, The Godfather.
You have two choices. Either work out your differences (much easier said than done), allow the relationship to be damaged in perpetuity by going your own way, or allow it to inflame even more. Incredibly, that is the stuff that create global wars.
So how do you stop that pendulum from swinging? First thing is to allow yourself some time to come back into balance. That might entail biting your lip a bit. Just don’t let time drag on too long, which could perpetuate resentment.
Then, once you are back into relative balance, what comes next next? You need to say something. It may just be just a few words, but let them know that the door is open to healing the relationship, to taking the next step. Be ready for the possibility of rejection in case they are not yet ready to take that next step. If so, just give them more time. Know that the peace offering will not be forgotten. Know that no one wants war. It is just that some people have a hard time taking that next step. You can always try again with another peace offering. Note that a peace offering is not assuming a fetal position. It is not ‘giving in’ or accepting defeat. It is a mature and adult stance, not a weak one. It may be challenging for you to make the offering, and they may not respond exactly how you would like them to respond. It is a process. Your goal is to steadfastly be a respectful, mature adult. Just keep in mind that the pendulum swings and it requires some skill to get out of its way.
When the time comes that both of you are willing to talk, know that neither of you will do that perfectly. Little jabs will be thrown. An ‘honest sharing’ of feelings may be more of a cloaked blaming than an honest airing. Just remember that and handle it as wisely as possible. The possible directions a conversation like this may go are too numerous to mention. However, the concept underlying all those possibilities is the same: stay out of the way of the swinging pendulum. Do not feed that swing. You may have to tolerate a hurtful jab here and there. If it starts to feel to abusive, you can respectfully excuse yourself from the conversation, saying you need to resume it later. More than likely they will know why and will reflect upon how they behaved.
This is an art. It requires patience and practice. A great deal of it is a function of your own personal style. There is no cookbook. Giving yourself the time to cultivate this art is much of what being an adult is all about.