Your employer is under no obligation to do anything here. That in unless required to do so under the terms of an employment contract or union agreement. The fact is that a company can set the conditions of the workplace much as it sees fit, absent some form of actionable discrimination. That having been said, your co-worker can be discharged for this if your employer so chooses. In an "at will" employment relationship, a worker can be discharged for any reason or no reason at all, with or without notice. That having been said, you could pursue action on your own against your ex-girfriend for "defamation", if what she has been saying about you is untrue, she knowingly has repeated rthese things to 3rd parties and it has caused damage to your reputation, etc. (although these cases are not always easy to prove and can be costly).