Penny de Valk's Answer:
It sounds as though the 'issues' you had before with that employee were unresolved and might explain her somewhat extreme reaction to what sounds like a reasonable request.
Perhaps she also shared her version of previous events with her colleagues. You therefore need to address this issue with your staff to understand what the real reason is behind their behaviour towards you and what behaviour of yours they have taken exception to.
Is there a staff meeting forum where you could raise the issue of what their problems are as you have noticed a deterioration in atmosphere since X's departure? They may tell you directly about things you could take quite personally, so try just to listen and not defend yourself.
If you get stone-walled ask why they think X is no longer there, they may tell you a version quite different to yours. Then you have an opportunity without any personal references to explain that there are standards of professional behaviour around patient care and between staff that cant be compromised and that is the only issue you had with X.
Don't get drawn into 'he said, she said'. Keep referring to professional standards of behaviour and ask them if they think they are fair and reasonable and whether it is unreasonable to expect everyone to manage their behaviour accordingly.
This may unearth misunderstandings that you will be able to help clarify and it may also unearth behaviours of your own which are having an effect on their motivation and goodwill that you might need to moderate.