Bergamasco, who has 84 international caps and is the elder brother of Mirco Bergamasco, has been replaced by Simone Favaro.
The 31-year-old is the third player to pull out of the Italy squad after prop Matias Aguero and flanker Josh Sole also withdrew through injury, with Franco Sbaraglini and Valerio Bernabo taking their places.
Italy fly out to South Africa on June 13 and take on world champions the Springboks in Witbank on June 19 and in East London on June 26.
Italy squad
Forwards: Valerio Bernabo (Rome), Marco Bortolami (Gloucester/ENG), Martin Castrogiovanni (Leicester/ENG), Lorenzo Cittadini (Treviso), Carlo Antonio Del Fava (Viadana), Simone Favaro (Parma), Paul Derbyshire (Petrarca), Quintin Geldenhuys (Viadana), Leonardo Ghiraldini (Treviso), Fabio Ongaro (Saracens/ENG), Sergio Parisse (Stade Francais/FRA, cap.), Salvatore Perugini (Bayonne/FRA), Franco Sbaraglini (Treviso), Manoa Vosawai (Parma), Alessandro Zanni (Treviso)
Backs: Mirco Bergamasco (Stade Francais/FRA), Riccardo Bocchino (Rovigo), Gonzalo Canale (Clermont/FRA), Craig Gower (Bayonne/FRA), Andrea Masi (Racing Metro/FRA), Luke McLean (Treviso), Simon Picone (Treviso), Matteo Pratichetti (Viadana), Kaine Robertson (Viadana), Michele Sepe (Viadana), Tito Tebaldi (Gran Parma)