Gibbs, who has suffered extensively from injuries in recent seasons, has not played since a 2-1 home win over Sunderland on October 16.
The latest fitness problem to afflict the left-back means he is unlikely to represent the club before Christmas.
"He has a hernia and had surgery on Thursday," Wenger said. "He looks like he will be out for between four and five weeks.
"At first he had an abdominal strain, certainly caused by the fact that he had a hernia already. When the strain was healed we discovered that he had a hernia. We had a fear that could be behind it so we decided to do the surgery."
"Carl has a stress fracture in his back," Wenger said. "That's just about rest now. We discovered that because he had a pain in his back. First you monitor it, then play a little bit and then you check again because you are cautious.
"We found a small stress fracture that we noticed at an early stage. If he continued to play it could have got worse and then he could be out for a long time. So at the moment the specialist has decided to rest him."
Abou Diaby and Marouane Chamakh have returned to full training but only the latter is in contention to play against Norwich City at the weekend. Japanese youngster Ryo Miyaichi is also out for "four to six weeks" with an ankle sprain.
Meanwhile, Arsenal midfielder Jack Wilshere has been given the all clear to shed his crutches for the first time since he underwent ankle surgery.
The teenager went under the knife two months ago after suffering a stress fracture in his right ankle during the summer.
He has not appeared for the Gunners yet this season, but is set to step up his recovery after he wrote on his Twitter page: "First day without crutches in a while! Feels so good!"
Wilshere had originally hoped the injury would heal itself with the help of a protective boot, however, after undergoing surgery he is due to return in the new year.