Levet broke his leg when he jumped into a lake to celebrate winning his home tournament on Sunday, while Clark has a long-term elbow injury.
The tournament at Royal St George's lost former world number one Tiger Woods on Tuesday when the American, who has been suffering from knee and Achilles tendon injuries, said he was not completely ready to return.
Levet suffered a small fracture to his shin when he leapt into the water minutes after winning the French Open in Saint Quentin last weekend.
"I am extremely disappointed to miss out on The Open Championship, but my specialist has advised me to have an operation on the fracture tomorrow morning," the 42-year-old Frenchman said on the European Tour website on Wednesday.
"I will have screws and a plate inserted which will ensure that my shin recovers completely.
"I will be off for six weeks which is very bad timing, with so many important tournaments coming up.
"However, the wonderful memory of winning my national Open will definitely keep me going through my recovery."
Last year's European Ryder Cup captain Colin Montgomerie strongly criticised Levet and the practice of players jumping into water to celebrate victories.
"I think it is the silliest thing that players have done over the years," he said at the Scottish Open.
"I have always been suspect about people diving into lakes, not knowing how deep they are and what is in there.
"There could be something sharp. He was lucky it was only a leg injury. It is not the way to celebrate -- taking the next three months off because you've hurt yourself."
Levet, who lost a five-hole playoff to Ernie Els at the 2002 Open, will be replaced in the field for the July 14-17 major championship by American Robert Garrigus.
Clark's place is now available to American Anthony Kim. Briton Simon Dyson and Dane Thomas Bjorn become the first and second reserves.