I think that it is very important that you get an acknowledgement of falsification of an abuse allegation. I agree with Renee though. Demanding an apology just seems self-centered but to demand that she openly admit to her wrong-doing is a protection from further harm. You have every right to demand that from her. (An admission of guilt, not an apology)
I made a demand during mediation that my lawyer told me was useless. I wanted my x to admit that a debt that caused me to have my paycheck garnished was 100% X's fault and that I had no idea the debt even existed. X, the mediator and my lawyer tried to argue me down, saying that there was nothing I could gain by it, but X gave in. I was merely demanding that the truth be acknowledged and the mediator could not argue with my point.
I have used that written statement to great advantage while seeking loans and other credit. We had the statement signed with the seal of a circuit judge. It saved my butt with creditors.
My point is that you never know what else this woman may try in the future, and you also also never know how the false allegation may come back to haunt you. Get this in writing while you can.
When my daughter admitted to making a false allegation against me I put that admission on my hard drive at home and at work and on two memory drives, plus many hard copies. If her lies ever come back to haunt me in the future, I am prepared to present evidence that it was a lie.
The fact that neither parent determines school and neither parent may leave the district makes you a completely equal parent at last.