Notice. I am assuming you wouldn’t have hired the employee back anyway and this was your justification?

Was she going to a competitor and you had concerns about the ability to gather information?

If that were the case then that is fine. If you are honest in your approach that you let her go because of the lack of a proper two week notice you are signaling that you really don’t care about individuals outside of them providing you some benefit. If that benefit ceases then their is no point for you to keep them around.

Why did she only give you a week notice? Maybe the pay was better but she had to start right away. Maybe she didn’t understand proper business etiquette. Maybe a conversation of, “that stinks to lose you but I understand your situation. I really needed some projects done, would you consider working some nights or weekends to help out?” If she really liked you as an employee she would have said yes if she could actually do it.

The feedback you are getting is that you drew a very hard one-sided line that indicates you are difficult to work with, you have a tendency to not care about individuals that don’t do it your way or have the emotional intelligence to understand why they would. And that action will be heard by other employees. Sounds like you have some work to do on your culture.

Feedback is a gift. Maybe you should consider it.