I was just accused of workplace bullying and harassment. It is untrue. What are my rights?

Do you have an employment contract or union agreement that affords you protection under the circumstances? Does company policy address this issue? If not, then as an "at will" worker, your employer can set the conditions of the workplace much as it sees fit (absent some form of actionable discrimination). In fact, you can be disciplined, up to and including termination, for any reason or no reason at all. As for the co-worker who is making these false claims, depending on the circumstances, you might be able to sue them for defamation. At this point, you may want to consult directly with a local employment law attorney who can best advise you further.

Unless you have a written employment contract, you  have no protection against any discipline, up to and including termination, your employer wishes to impose, whether fair or not, or even whether factually based or not: without a contract, you are an "employee at will" and may be terminated (or otherwise disciplined) at any time, for even reason, even due to supported allegations.
If you can identify whomever is making these allegations against you, if they are, as you indicate, untrue factual claims (not mere opions; it must be a false factual statement), you could sue that person for defamation; but the employer is not liable for an employee's wrongful defamatory act in this regard, so you could only sue the person making the fales statements about you.