The answer is, it depends.
This situation actually seems to be occurring more and more due to recent media stories. Companies hear about something, want nothing to do with it, and will terminate someone to make it go away. Guilty until proven innocent is really how it works now. Person C definitely has a case if he/she was wrongfully terminated, but it also depends how far they want to take it. Was this issue made known? Will it follow them around and hurt potential job leads? If the person was wrongfully terminated C can also go after A for claiming false witness. This of course depends on B's willingness to tell the truth and how involved they were in the original "investigation" (they might be on the hook for their job too).
I have a family member which this same thing happened to (they were C). It has been a long, painful, and expensive process. People will lie and do whatever it takes to save their job even if it seems obvious how guilty they are. Had there not been inaccurate media coverage tarnishing this family member's reputation, they probably would have just moved on with their life and got another job.
It also depends if the company is an "at will" employer. In this case there is very little that can be done, since they can terminate you for no reason at all.