As a Christian, if you recognise you said something which was wrong, you ought to apologize regardless of who started this verbal fight.
You need to recognise that many people (mostly unsaved people) will quickly go very cold and distant after a verbal fight. There is also the issue of pride as to why they will not apologize first. I would not automatically assume that this other person does not care about the relationship - it may be pride or something holding them back.
If you do the right thing by apologizing then the person may in turn apologize to you for starting the whole thing. It's a good testimony to be the one to make the first move.
Matthew 5:23-34
"Therefore, if you are offering your gift at the altar and there remember that your brother has something against you, leave your gift there in front of the altar. First go and be reconciled to your brother; then come and offer your gift.
Luke 6:27-28
But I tell you who hear me: Love your enemies, do good to those who hate you, bless those who curse you, pray for those who mistreat you.
Luke 6:35-36
But love your enemies, do good to them, and lend to them without expecting to get anything back. Then your reward will be great, and you will be sons of the Most High, because he is kind to the ungrateful and wicked. Be merciful, just as your Father is merciful.
Keep in mind these scriptures for the future as we all have to try and get into verbal fights, it is easy to return a tooth for a tooth and an eye for an eye but Jesus calls us to keep the peace:
Romans 12:17
Never pay back evil for evil to anyone. Respect what is right in the sight of all men.If possible, so far as it depends on you, be at peace with all men.
Luke 6:31
Do to others as you would have them do to you.
Proverbs 15:18
A hot-tempered man stirs up dissension, but a patient man calms a quarrel.
Ephesians 4:26
"In your anger do not sin": Do not let the sun go down while you are still angry,
And remember, if she does apologize, show mercy and forgive her.
Matthew 6:14
For if you forgive men when they sin against you, your heavenly Father will also forgive you.
If your apology is not accepted then simply leave it there, you have done what was right before the Lord, if your friend is unforgiving then she is then 100% wrong, but never the less, keep your door opened, she may come round and seek your forgiveness then in turn you can show the mercy she never showed you.
Matthew 5:16
In the same way, let your light shine before men, that they may see your good deeds and praise your Father in heaven.