Your right to privacy, or your right to your boss not talking about you to other employees is limited. It is not inappropriate for your boss to investigate issues related to workplace issues. A boss is allowed to ask other employees details of the workplace, including issues related to why a cash register is short money, whether a certain employee is goofing off, talking on their cell phone while at work, miss-marking prices on products etc. An employer can look into any issue that effects the Company's bottom-line. If you are caught in the cross-hairs of the investigation because it is your till that is short money, it is simply an unfortunate result.
Of course, there is a professional way to handle and investigation or issue and an unprofessional manner. However, a boss that does not handle investigation professionally still has not likely crossed the line into an illegal realm.
If the employee making these statements has no authority greater then you-- this may be an issue you need to address with a supervisor, especially if it is not true. Idle gossip can lead to bad consequences in the workplace.