Castrogiovanni has been ruled out for at least a month following tests by Italy's medical department but has an outside chance of making the Azzurri's final two fixtures away to Wales on March 10 and on home soil against Scotland a week later.
"The initial diagnosis is four to six weeks, and that's bad luck for Castro," said Cockerill.
"He was really up for the England game and carried the ball a lot. But then he fell on the ball, got a bang in the ribs and he has cracked one."
Meanwhile former Azzurri skipper Marco Bortolami, who started in the second row in the defeat to England, insists Castrogiovanni's injury opens the door for the promising Lorenzo Cittadini who replaced him against the Red Rose.
The Treviso tighthead has impressed at club level, and Bortolami believes he is already the match of his more experienced colleague at scrum time, while he is working to play as big a role around the park.
"Lorenzo has a chance to prove he is probably even better than Castro in the scrum," said Bortolami.
"The team has a lot of confidence in him, and he will get the opportunity to show his abilities in the next few games.
"Of course Castro is a huge part of the team, but we are very fortunate to have someone like Lorenzo to come in."