Dear Sarah,
I am writing because I have been feeling terrible about our argument last Tuesday, and I wanted to reach out to clear the air. I am truly sorry for my part in that misunderstanding; I reacted far too defensively and did not give you the chance to explain yourself properly.
From my perspective, I felt blindsided when you mentioned my lack of communication regarding the weekend plans. Because I had been under immense pressure at work, I misinterpreted your concern as a personal criticism of my reliability rather than just a simple request for information. I was already stressed, and I projected that frustration onto you, which was completely unfair.
I really value our friendship and hate that things are awkward between us. Perhaps we could meet for a coffee this weekend? I would love to talk things through calmly and properly listen to your side of the story. I promise to be a better listener this time so we can put this behind us.
Let me know if you are free on Saturday.
Best wishes,
Mark