Sickness during annual leave

A member of staff booked 2 weeks holiday, which was agreed by the company and her first day of her holiday was the Monday.  On the first Wednesday she rang in to say that she been taken ill, and as she had not arranged to actually go away on holiday, she now wanted to take it as sick leave and have her holiday at a later date.  I asked her to supply me with a fit to work certificate from her doctor, which she said she would.  She came back to work, having taken the two weeks off, saying that as she had only been ill for 3 days, she carried on with her holiday and so only wanted to take the 3 days at a later time.  She had not been to see her doctor and supplied me with a self certification note saying that she had an upset tummy.

I told her that as I did not have an actual fit to work note from her doctor I had no proof that she had actually been ill and so she would not be able to have her 3 days later in the year.  I have searched the Internet for information and have found the following statement "Until the European or UK courts say otherwise, our view is that employers are entititled to require workers to produce convincing evidence of their illness while on holiday".  I do not feel that a self certification note is convincing evidence.

Was I correct in saying what I said?

Thank you.