Ogilvy shot a three-over-par 75 in Thursday's opening round at the TPC Sawgrass and then carded a two-over-par 38 on the back nine Friday before pulling out with a shoulder injury.
The 2006 U.S. Open winner had been struggling with the injury and had missed last week's event at Quail Hollow due to the problem.
Clark, who also shot a two-over-par 42 after 10 holes on Friday following a two-over-par 74 on Thursday, had also been sidelined by a nagging elbow problem that had kept him out of action since last month's Masters.
Earlier this week, Clark had said he was not going to let the injury get in the way of defending his title at the Players Championship. But he began to experience pain Thursday night and again Friday morning.
"It has just been getting worse as the day has gone on. There's just no point in staying," said Clark, who hopes to be back in a month or two at full strength. "It's a long year and I wouldn't want it to end right here, so it's the best move."
The pair follow Tiger Woods out of the tournament after he withdrew Thursday with knee and Achilles tendon injuries.
(Editing by Frank Pingue)