Scalise received numerous blood transfusions as surgeons worked to fix his torn blood vessels and other wounds, Sava said, adding: "We are encouraged by improvement in his condition over the last 36 hours".
When Scalise arrived in the hospital, he had an "imminent risk of death" and was "as critical as you can be", doctors also said.
David Bailey, one of the Capitol Police officers injured in the shooting at a baseball field in Alexandria, Virginia, threw out the first pitch at the game. According to the hospital, he sustained a single shot to his left hip, with the bullet traveling across his pelvis causing bone fractures, internal organ injury and severe bleeding.
Nonetheless, said Sava, "We fully expect him to be able to walk" and "hopefully run".
More on this as it develops.
But Scalise can hope to make an "excellent recovery", his trauma surgeon said Friday.
The wife of wounded GOP congressman Steve Scalise says she is thankful for all the prayers and warm wishes her family has received since her husband was shot at a baseball practice.
The American Hospital Association defines "critical" condition as: "Vital signs are unstable and not within normal limits".
"In Washington we have our disagreements, but we all agree that we are here to serve this nation we love, and the people who call it home", President Trump said in a video address to the crowd.
Republicans and Democrats are fighting for bragging rights in the annual congressional baseball game as one unit - Team Scalise.
"The security detail saved a lot of lives", he said. Scalise was hit as a gunman sprayed bullets into a GOP congressional baseball practice outside Washington D.C.
The president's meaning was unclear as he made the comments at an event on Cuba policy in Miami.
Ms Griner remains in hospitali at MedStar after being shot in the ankle and Mr Bailey was spotted on Friday in the Capitol, on crutches and out of uniform, accepting congratulations from fellow officers.
House Speaker Paul Ryan has paid a visit to wounded Majority Whip Steve Scalise at the hospital.
This year's congressional charity baseball game is proceeding after a rifle-wielding critic of President Donald Trump opened fire June 14, 2017 on U.S. lawmakers practicing for the charity event, seriously wounding a top Republican congressman and three others before he was killed by police. The offices of both senators - Dick Durbin and Tammy Duckworth - turned over the emails to Capitol Police and declined to offer specifics about their content. He's been undergoing repeat surgeries.
The Washington Nationals released additional seating in Nats Park to accommodate the demand.
Colleagues said Scalise, who had been fielding balls at second base, dragged himself away from the infield, leaving a trail of blood before they rushed to his assistance.