While Bhaskar has provided a solution on how to handle the problem at hand, I would like to delve a little deeper.
Interpersonal Environment and Feedback Culture
Is this a sign of the interpersonal environment in your company? Can it be construed that there is a lack of a culture of feedback in the company? Imagine for a moment, whatever disgruntlement that female employee had, if she had expressed it in a professional manner by having a tête-à-tête with her colleague, would it not have strengthened the bond of friendship? Would it not have kept the couple's motivation level the same? How come that female employee did not bother about the consequences of her revilement?
These are not major incidents, but they are harbingers of conflict, groupism, lack of teamwork, etc. The story could be making news or will start making news in the grapevine after a couple of weeks or months. This informal communication shapes the organisation's culture. Therefore, I request you to look at the larger picture as well.
Training and Action
The incident calls for training your staff on office decorum, how to give feedback, how to accept feedback, interpersonal skills, etc. If you have conducted training on all these subjects, then should it be construed that training efforts have failed? Nevertheless, even after the training, if this incident has happened, then the incident merits strong action against the erring employee. This will send a strong message to one and all that training needs to be taken seriously. Training is conducted for implementing the learning in day-to-day activities and not to be abandoned after stepping out of the training hall.
Whether the badmouthing was on a personal or official count, tell the aggrieved employee to submit a written application. That is the starting point for your action.
Thanks,
Dinesh V Divekar