As a business owner, if a customer has a gripe with an employee, it is my or my manager on duty's job to straighten out the issue and make the customer happy so they continue to do business and provide positive word of mouth.
If a customer demands that I fire an employee because they don't like the way they were treated, I would consider that an unreasonable demand and that customer may remain dissatisfied in that respect. However, a serious discussion with the employee would take place after the situation is defused.
If I tell the customer I will fire the employee, I had better have good grounds to do so or the employee will have a claim against me. I have dismissed two employees for incompetence and other issues. One went to court. I prevailed. In any case, the employees were fired, and remain fired. They are both customers now, as they understand that they were treated fairly and that their dismissal was not a personal issue, but professional.
If I did not intend to fire the employee, I would explain to the customer that the employee will be reassigned to other duties pending completion of retraining/review/anger management/education or whatever it takes to get the employee ready to carry out the assigned duties in a professional manner, and would then do so. And I would likely personally handle that customer's calls for a while to re-establish good will.
Vikki 1969 Goldenrod Yellow / black 400 convertible numbers matching