Healy withdrew from the starting XV for last weekend's 40-7 rout of Northampton and had already been ruled out of the return fixture at the Aviva Stadium on Saturday.
However, it was decided he required surgery on his right ankle and the British and Irish Lions loosehead is facing an eight-week spell on the sidelines.
He will miss the return against Northampton and also the remaining Pool One games against Castres and the Ospreys.
"Cian had ankle surgery on Wednesday so he's unavailable for about eight weeks. He's doing well but he won't play for a while," Leinster coach Matt O'Connor said.
Leinster took full command of Pool One after demolishing Northampton, leaving then five points clear at the top of the group, but O'Connor has warned against complacency.
"In the Heineken Cup you must make sure you put out the best quality performance week on week," said the Australian, who is hoping flanker Sean O'Brien recovers from a dead leg.
"We're under no illusions how tough next week will be in front of a big crowd at the Aviva Stadium. We must make sure we're as good again."
The Ospreys are hoping to record their first win in the competition this season when they host Castres at the Liberty Stadium on Friday night.
Many had tipped Steve Tandy's side as the most likely Welsh region to deliver a strong Heineken Cup campaign, yet they are as good as out following three straight defeats.
Cardiff Blues director of rugby Phil Davies has warned his side they must raise their game to a new level on Friday night if they are to complete the Heineken Cup double over Glasgow.
The Blues head north to face the Warriors at Scotstoun having emerged 29-20 winners at the Arms Park when the sides met last weekend.
"We have been getting better all year and at the moment the performance levels are growing," Davies said.