I falsely accused my co-worker for a mistake that someone on my team made.
Do you think they deserve an apology?
I don't want to make them feel bad about the situation. Also if I had a genuine mistake I would apologise
You should never make accusations without having all the facts first. You should apologize to them regardless.
No it's fine and I wouldn't feel sorry if they are accusing me of making false accusations.
Maybe you mistakenly you accused your coworker but still you need to apology for that. Your insensitivity will cause tensions in your working relationship and ultimately will affect your work. So in order to keep the work environment healthy you need to say sorry to them.