You should apologize.

Not because you were wrong (although maybe you were)
Not because it will fix things (although it might help)
Not because you will gain anything from doing so (although you might feel better)

You should apologize because someone is upset.

A friend's well-being is more important than your justification.
Peace that keeps communication possible is more important than what you know.
A friend feeling loved is more important than your misunderstood intentions.

Even if you were right
Even if they misunderstood
Even if they overreacted

Even if it's unfair.

My students display human nature more clearly than any of my adult friends. By our interactions, I've learned so much about apologies. When I am appalled by their excuses, their instinct to justify their actions in the presence of a hurt peer, I am convicted.

Though we recognize it easier in children, it's what we all do.
Our first instinct when accused is to explain ourselves- but I don't think it should be.

When a loved one is upset, shouldn't our reaction be compassion? Shouldn't our aim be to keep lines of communication open? Shouldn't we crave our loved one's comfort?

Even if those require a sacrifice?

Maybe it was a misunderstanding. Maybe you were right. Maybe they have been unfair to you.

But unless they know you're concerned about them- the communication that needs to take place won't happen.

So leave your excuses behind and apologize.
If there have been hurt feelings, I guarantee you there is something to apologize for.