Falsified writeup

I received a corrective action writeup on 3/24/24, for actions that I didn't do. I tried telling the manager that these were false accusations and she refused to hear my version and said that she only wanted my signature on the line. (Which I didn't give). I asked where the verbal written warning notification was at and she said that it wasn't necessary to do. I have since filed a grievance through sn outside company for HR to investigate but feel like they really didn't do anything to the coworker that started the false accusations and negative rumors about me in the dept. I also submitted a formal rebuttal to HR; who has recently said they did their investigation and they agreed with the manager for the writeup. I requested that video footage be pulled, badge taps, interviewing several other employees that were present; but I have a feeling they didn't do anything except wait 3 weeks to answer. I want to request a copy of the investigation from HR, is this OK to do? The coworker that has started everything in Dec and it escalated this far, has been allowed to create a hostile workplace for others previously and was caught giving away drugs from the pyxis to patients for free under other patient names. She didn't get any punishment for creating false information in a writeup. I'm just at a loss of what to do next.