The first thing I’d say is that I see no hope of this situation ever changing as long as you both refuse to apologize. And that’s tragic, because eventually one of you will die and then it will be too late—and whoever is left will live with remorse and regret the rest of their days.
After all, what keeps you from apologizing? Perhaps you feel that your sister, and she alone, was in the wrong. But even if that’s true, your estrangement has gone on far too long, and frankly, it almost doesn’t matter now who was responsible. Jesus said that if we have something against someone, we should “First go and be reconciled to them” (Matthew 5:24).
However, I suspect an even more serious issue is causing this barrier, and that is pride. None of us ever likes to admit we’re wrong; it’s part of our sinful human nature. But when pride gets in the way, it needs to be faced and confessed to God. When Jesus was asked how often a person ought to forgive someone who’d offended them, He replied, “Not seven times, but seventy-seven times”—in other words, endlessly (Matthew 18:22).
Turn to Jesus Christ and ask Him to forgive all your sins, including your attitude toward your sister. Then ask Him to help you reach out in love, and do everything you can to bridge the gap. Years from now, you’ll be glad you did.