A security guard saw the man touching the statue and tried to stop him but was too late to avert the damage.
The sculpture, believed to be the work of Giovanni d’Ambrogio, was on view at the Museo dell’Opera del Duomo.
It is unclear how much it will cost to fix the finger, which was not original and had been added at a later date.
The tourist, who is said to have apologised, could receive a fine and has been publicly chastised by the museum’s director.
“Do not touch”, said Monsignor Timothy Verdun, was a “fundamental” rule of museum-going that had been forgotten.