My sister, who lives in another state, refuses to talk with me. I have no idea what I did and so I have no clue how to handle this. Should I apologize in the hope that this will soften her heart so that she will at least talk with me?

It is obvious that you see no unjust behavior on your part. When we seek forgiveness from others, it is in the context of knowing that we have been unjust. Thus, it follows that you should not ask for forgiveness for something that you did not do. With that said, it is reasonable to acknowledge your sister’s hurt feelings. A way to acknowledge this is to say something like this to your sister: “I am sorry if some of my behavior has caused you pain. Can we talk about it?” Notice that you are not acknowledging wrong-doing (because there was none as far as you know). Instead, you are acknowledging your sister’s hurt feelings, a situation you would like to address.