If you're sincerely sorry for what you did, then you certainly should apologize to her -- even if you think she'll reject your apology. But you shouldn't only apologize; you should also ask for her forgiveness.
After all, if you don't seek forgiveness, the split between you will always remain. More than that, you'll always wonder if you might have misread the situation, and therefore missed an opportunity to heal the hurt you caused. From what you say, you're the one at fault for what happened, and therefore it's up to you to take the initiative and reach out. The Bible tells us to "Live in peace with each other" (1 Thessalonians 5:13).
We can't control how people will react when we've wronged them and then ask for forgiveness; they may or may not accept our forgiveness. But that shouldn't keep us from facing honestly the wrongs we've done, then doing everything we can to mend the damage. Remember: Not everyone accepts God's offer of forgiveness, but He still offers it to us. Have you turned to Jesus Christ for the forgiveness you need?
In addition, learn from this experience and resolve, with God's help, to avoid hurting others in the future (both by what you do and what you say). Ask God to help you to be kind and compassionate toward others. The Bible says, "Do nothing out of selfish ambition or vain conceit, but in humility consider others better than yourselves" (Philippians 2:3).