Well you've gone and done it now. You're not alone. Just kiddin', kid. We'll figure this out.
I'm sure you didn't mean to piss them off. It happens.
Look, I'm not here to say you that you were wrong or you were right. We've got ourself a situation we've got to take care of.
It doesn't matter who was right at this point. First, acknowledge their anger. Even if you think they are being irrational. Hear them and really listen to them. Maybe they have a good reason to be pissed.
How can you fix this and move on? Can you come up with a solution that will make both of you happy? See what it will take to solve this. They need your help with something.
Side note: Is this client beyond Pissed Pissoffterson and threatening you? Check your contract and see what the options are to ending the relationship. You can't cure crazy.
Once you have a solution, take care of it right away. Let your client know that you're on the motherfucker and you will keep them updated.
Just do your best. That's all you can do.
p.s. when it's solved and you've moved on, go back and see how you can learn from it. That's the way you'll grow.