For Christmas, I would definitely show up. Go there, be the most well-adjusted person they've ever met, and show them that their fears were unfounded. Be helpful, courteous, polite... basically schmooze their pants off.
In other words, win them to your cause.
Depending on your relationship with them, maybe you can even go over there sooner than Christmas and ask their advice. Explain things, calmly and thoughtfully, from your perspective, without any implications that the other girl is trying to be a home wrecker, but only relaying the facts of your interaction (she turned her back when I tried to give her a hug, she won't even look at me when we're across the table together) without your personal spin on them.
Express how you regret writing such an angry letter and that you didn't mean to hurt anyone, that you'd been bottling up your pain and jealousy and that you probably could have released your emotions in a more constructive way. At this point, you need to focus on damage control.
You have every right to feel the way you did and it does seem like your boyfriend has been playing some nasty games. Now if it were me, I would just chalk him up as dysfunctional and move on. Some people are beyond repair.
Since you seem determined to save the relationship, despite his demonstrated lack of polyskillz or consideration of your emotions, you'll basically need to do the work for everyone. I don't even know how to begin to advise on how to do that, because it's not a game I play. I have a very low tolerance for bullshit...
He's saying you won't change. If you want to save the relationship, you'll have to ask what changes he wants you to make, and you'll have to make them. I personally disagree with making people change, but he seems set on making you change, and you seem set on staying with him, so I guess that means you'll have to change, right?
I'm trying to not-so-subtly hint at the possibility that staying with him may not be not in your best interest. I understand that you've invested time with him, but if he's hurting you like this, you need to ask yourself: is it worth investing more time? Or might it be better to invest that time and energy in yourself, and find someone who won't make you be someone you're not?
“As I am sure any cat owner will be able to tell you,
someone else putting you in a box is entirely different
from getting into a box yourself.” —bisexualbaker