Am I at fault here or what?

Although you may have started the argument (and totally over reacted to her taking a raincheck) she totally made a rain puddle seem like the pacific ocean. If you apologize again in a bit, say wait a week or two, she may have cooled down from lava hot to microwave hot. Just wait. And if you wait more than a month and she still won't forgive you, that means she isn't worth it, she doesn't want your friendship, and you're going to look totally hopeless and desperate. Give it just a little while