Two secret videos showing rapid deterioration in Sgt. Bowe Bergdahl's health persuaded reluctant military and intelligence leaders to back the prisoner swap that has stoked a backlash, officials said Tuesday, as the Army launched a new probe into why the soldier disappeared from base shortly before his capture by the Taliban in 2009.

The decision that brought about Sgt. Bergdahl's freedom on Saturday continues to draw criticism. Sen. Dianne Feinstein (D., Calif.), head of the Senate Intelligence Committee, said Tuesday that Congress should have been consulted before Sgt. Bergdahl was exchanged for five Taliban detainees, who were sent to Qatar.

President Barack Obama, speaking in Warsaw on Tuesday, said questions about Sgt. Bergdahl's disappearance—which some soldiers said they believe amounted to desertion—will be answered. "Regardless of the circumstances, whatever those circumstances may turn out to be, we still get an American soldier back if he's held in captivity," Mr. Obama said. "Period. Full stop. We don't condition that."

Administration officials sought to tamp down anger among lawmakers by offering a private briefing on Tuesday. But Ms. Feinstein said she heard no evidence that Sgt. Bergdahl was in immediate medical danger that made it necessary to act without consulting Congress.

"There certainly was time to pick up the phone and call and say 'I know you all had concerns about this, we consulted in the past, we want you to know we have reviewed these negotiations,' " she said, adding that a White House official called to apologize for not providing more notice to Congress.