Employees sometimes use subterfuges for getting the time they want off at Christmas and one is to claim illness.
It is entirely appropriate to challenge an employee about the nature of an illness at Christmas, and especially so if they have asked for, and been refused time off. But it is best to start from the premise that the illness was genuine and ask questions from that standpoint. Although it can be difficult to assume genuineness in some cases, there is nonetheless a lot of flu about; so an illness might well be genuine.
Ask the employee when the illness started, what were the symptoms, was s/he bed ridden, how did s/he cope with the sickness, what did s/he take for it? What does s/he think caused it? Who else in the family was ill? If the illness is a deception, then help the employee to weave a tangled web. It may be difficult for them to fully maintain the deception.
Make the employee feel uncomfortable about the event, and especially so if you suspect a deception. Even if genuine, sickness when they asked for time off doesn’t look good for them.
If questionable illness happens again at a later date, then you are building up a picture and you may be able to take action then.
But unless you have indisputable evidence that the illness was non-genuine (for example, the employee was partying at the very time that they did not come to work), then it would be unwise to contemplate a warning on the basis of a single incident.