Investigators say they still do not know if the shooter James T. Hodgkinson had an attack plan. This person was not authorised to speak publicly about the case and spoke on condition of anonymity.
President Donald Trump, who visited Scalise in the hospital on Wednesday, sent a video message that played at the start of the game.
He is now in serious condition, an improvement from the critical condition he was in since his first surgery on Wednesday.
Sava declined to put a timeline on when that would happen or when Scalise, 51, would be able to leave hospital. Sava called Scalise's injury a "trans-pelvic gunshot wound", describing how a bullet entered his body near his left hip and traveled toward his right hip, causing "substantial damage to bones, organs and blood vessels".
The No. 3 House Republican remains in intensive care and is likely to stay hospitalized for weeks.
A Republican supporter holds up a sign supporting US House Majority Whip Steve Scalise.
The Louisiana Republican is "more responsive, and is speaking with his loved ones", according to a statement posted on his Twitter account.
He has had several operations since the shooting.
His sedation level has been lowered enough to where he has been able to communicate with doctors and his family. Capitol Police said they had "no evidence to suggest that the purchases were not lawful".
Rep. Steve Scalise, R-La., and four others were injured in the shooting before U.S. Capitol Police fatally shot Hodgkinson.
Hodgkinson, a Belleville, Illinois, home inspector who had been living out of his van near the park, had a social media page filled with criticism of Republicans and the Trump administration.
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".
Once he has recovered, Sava said, Scalise will require rehabilitation.
"I feel a lot more confident and a lot more optimistic than I did two, three days ago", Dr Sava said. The congressman's risk of death was now substantially lower because doctors have controlled the bleeding and his vital signs have stabilized.