The Lyon midfielder, 23, was in tears when he limped out of training at the national team's pre-tournament camp at Clairefontaine late on Thursday afternoon, and his worst fears were confirmed by further medical examination on Friday.
Grenier struggled with an adductor problem throughout last season, and missed three months of the Ligue 1 campaign due to the injury and a bacterial infection contracted during treatment.
Grenier, an OL youth academy product, had appeared to be over the problem having appeared in his club's last three league games of the campaign, and was effective after coming off the bench in Les Bleus' 1-1 draw with Paraguay last Sunday.
Grenier's withdrawal means Cabella, 24, will get the opportunity to add to his sole international appearance to date, which came in the 4-0 win over Norway at the Stade de France in his nation's first World Cup warm-up game late last month.
The Montpellier man, who was reportedly poised to join Newcastle before they balked at the Ligue 1 club's 15 million euro price tag, was preferred to Southampton's Morgan Schneiderlin, who is seen as a more defensive-minded midfielder.
Meanwhile, Olivier Giroud believes a united France squad can go some way to making amends for their poor showing four years ago in South Africa.
In the 2010 World Cup, France failed to make the knockout stages of the tournament after finishing bottom of their group and made headlines when players boycotting training after Nicolas Anelka was sent home for verbally abusing then-coach Raymond Domenech.
"I am looking forward to the World Cup a lot," he told Arsenal Player. "It will be amazing in Brazil and a really good opportunity for France to bounce back. "We had a big disappointment at the last one but it is part of the past. Now we have a really good squad so we have a good chance.
"Yes [the fans still talk about the 1998 World Cup] because for the French people that will always be a fantastic memory. They're expecting a lot from us.
"We have shown them that we are better than one or two years ago. We have really reconciled with them because of our good results.
"We had bad memories from the World Cup and the last Euros, but they are behind us and 1998 is the past -- you have to look forward. We have the support of the French people, so that is a good thing for us."
Deschamps hopeful of Ribery return
Franck Ribery is in a race to be fit for the start of the FIFA World Cup and avoid the growing number of injured absentees from the tournament.