Capuozzo suffered the injury early in the 31-29 win over Scotland last weekend, as he attempted, unsuccessfully, to stop Kyle Steyn from going over for a try, but the fullback remained on the field for the remainder of the game in Rome.
"In the first minutes of our beautiful, rough win against Scotland, I injured my finger. Tests carried out since then showed a displaced fracture," Capuozzo said on Instagram.
"I tried everything to finish the tournament with the team against Wales, but I was forced to stop for emergency surgery."
Capuozzo said that the operation went well and he hopes to return as soon as possible with his club Toulouse to finish the season.
He missed the opening Six Nations game against England through illness but has played the three games since then, scoring a try in Italy's 13-13 draw with France.
Lorenzo Pani replaces him at full-back for Saturday's game in Cardiff.
