Should I write a letter apologizing, or leave it alone?

My friend was forbidden to talk to me, and I really miss her. So I was thinking of writing a letter to her mother explaining why I am sorry, though technically it wasn't my fault. What happened was I told her I had mono and I could die from it. I was diagnosed by a physician and everything, and if your spleen swells up, and a force makes it rupture, you could die. She started crying. Her mom called me a drama queen and said that you could die from a splinter, but you couldnt die from mono. I told my mom, and my mom got furious, and said we couldn't talk. I called her one last time, and she started crying. Her mom said I made her an emotional wreck that past week, and she needed to hang up the phone. A week later, when I tried to call her because I could talk again, her mom won't let me talk to her. I want to write a letter apologizing, but then again I feel as if it's not my fault. But I really want to talk to her, and I don't know what to do.