The England international underwent the operation on Tuesday after visiting specialist to find out the full extent of the problem which has been troubling him since September.
The 38-year-old shot-stopper, who has missed Arsenal's last 13 games plus England's games against Greece and Sweden, is set to be out for a further six weeks and Gunners boss Arsene Wenger is refusing to put a date on his comeback.
"David will be out for six weeks, two weeks to recover from the operation and four more to get completely match-fit," said Wenger.
"He has a calcification on a bone in his shoulder and it had to be cleaned out. We have already been without him for so long and although I didn't expect it to come to surgery we have had to get it sorted out."
It leaves the Gunners with just third-choice keeper Stuart Taylor for next week's Champions League showdown with Juventus as summer signing Richard Wright is still missing with a knee injury picked up in last week's defeat to Deportivo La Coruna.
Wenger is hopeful that Wright will be back in ten days and that he will not have to bring in anyone on loan as Taylor will continue to deputise.
"He will play against Ipswich on Saturday and Juventus on Saturday, but if Wright is back in ten days as we expect we won't have to think about buying or borrowing another goalkeeper," said Wenger.
"In any case a newcomer now would not be eligible for the Champions League until February."
