Question posed: is it seemly for a manager to apologize to someone he supervises?
Answer: It happens, as an intervention proceeds, that parties in conflict learn more about the other parties’ perspectives. Often, the result is that someone wants to apologize for behaviour that seemed reasonable at the time. Learning from the discussion in the mediation what the impact of that behaviour was on the other people, can put that behaviour into a whole new light. Has that happened in this case?
It sounds like you are concerned about losing face, or diminishing your authority in the employee’s eyes. It’s a concern based, in part, on a belief that power comes from being strong and always correct. Is that concern true or an assumption based on a blind spot?
It’s possible to share your insight as a different way of managing. In this example, you aren’t giving up any power; your authority remains unchallenged. What you offer is to learn from the communication you would henceforth have with each other. The employee might be very happy with the outcome and you both feel empowered with your new knowledge and conflict competence skills.
What is the worst that can happen if you tell the employee you’re sorry for how you acted, and you hope they can repair the relationship and continue to work together with more 2-way feedback than you’ve had before? If that worst scenario were to occur, what would you do next? And next after that? If you have ideas for learning more skills, both you and the employee are already ahead of where you were.