I’m pretty sure that’s not right, they cannot require a doctor’s note unless it’s over a certain number of days off. If it’s just the one day it’s unreasonable to require a note.
As long as there is no consistent pattern of absences they should just accept that you are ill and leave it at that, (although if they’re feeling particularly petty, they can insist on a return to work interview). They are absolutely not supposed to ask for a doctor’s note for a single day off because it’s an enormous waste of everyone’s time, and of course if you are ill, you shouldn’t be moving around trying to get a doctor’s note.