Maintaining contact to get her side of the story is mad. She was fired for gross misconduct because she abused a service user. It is not your place to establish the facts and go meddling. That extends to gossiping about it amongst colleagues as well. It would be unprofessional to reach out to her to hear her out, from the company's perspective you would be contacting a known abuser whose victim you care for, so you could get yourself dragged into this mess and affect your own career by doing so.
There was CCTV of the incident. Therefore there should be solid evidence of what has happened and it cannot be misconstrued. If still, it's all a big fix and she was perfectly justified in her actions, then she is within her rights to appeal and I'm sure if she wants to clear her name she will do so. Even then, if she finds the process has not been fair, she can take this to tribunal. And with CCTV evidence of it all, this would truly clear her name if she was wrongly fired.
Even if you speak to her, and she gives you a good story. What can you do with that exactly? Lots of abusers will twist it around to make the victim look responsible. So you won't even know if it's true or not. Pointless exercise and too much risk.
Also, in my experience of care, if you're a bit snippy on a one off because you've been pushed to the limits, this would not lead to gross misconduct. Perhaps you would get a warning, some extra training, but I very much doubt you would be fired for it. We are human after all and sometimes mistakes do happen. But abuse is not a mistake. As a care worker, I have never been abusive verbal or otherwise because I've had a hard day, and trust me I've had plenty of hard days! And as hard as it is, you do get to go home at the end of it, even if you're doing back to back sleep-ins. So it isn't really comparable to people who have to deal with it relentlessly in their own home with their relatives.
Furthermore, I'd be disgusted if someone reached out to a member of staff who sexually abused someone in their care to hear their side of the story. So let's not do that in other cases of abuse whether there is a chance of them being untrue or not.
I am not saying you will do any of this OP, but I am shocked at people suggesting to find out the other side of the story.