If you know the names of the other employees who have complained, you may want to first check with them to see what happened, if anything, in their cases. Talking with them may generate additional names you were not aware of.
Sometimes HR does address issues privately with the person complained about (PCA), while the complainant never knows exactly what happened. This is because the PCA was counseled, coached, had a note to their file, received remedial training, etc. but still remains in his/her job. However, you specify that after complaints by multiple employees the manager's behavior hasn't changed.
Upon contacting other employees, it would be most helpful to you in carrying the issue forward if you could make a list regarding the following:
1) who complained and their demographics, if you suspect demographics is an issue (e.g., gender, race)
2) what they complained about
3) approximately when they complained
4) which HR employee handled the case, and
5) what the result of the complaint was.
You may not know all of this information, but try to fill in what you can. (Some coworkers may not want to be involved.) Next, look for trends. For example, if complainants have been all females or if all employees were complaining of being yelled at and accused of doing something they didn't do, then note the trend.
Print out a copy of the email in which the manager falsely accused you of something. Print out copies of any other relevant documents and outline your overarching complaint (covering the group of employees). The more organized and logical you are in your presentation of your complaint, the more convincing a case you'll be able to convey.
Next, make an appointment with the Director of HR and present the information. If you've already talked with him or her (or they are part of the problem), then go up the organizational ladder. If you don't like that option, you may also complain to your boss' boss and go up the chain there, presenting the information you compiled.
It is most helpful if a group of several employees meet to complain about the lack of action. In the meeting, make sure you know what solution you're looking for. Do you want your boss retrained, demoted, fired?
If it were me, I'd team up with prior complainants and lodge a group complaint against HR for failing to adequately respond to a trend of managerial misconduct. There is power in numbers. Good luck.