"The Scalise family greatly appreciates the outpouring of thoughts and prayers", the statement concludes.
"Scalise continues to show signs of improvement". Doctors immediately performed surgery, but additional operations will be required, the hospital said in the late-night statement.
Sava said Scalise will require surgeries soon for abdominal and bone injuries, in addition to the two he has undergone to fix a bone in his leg broken by the bullet.
Investigators are trying to determine what his intent may have been - whether it was a list of potential targets for violence or whether he wrote them down for some other reason, one official said.
Scalise can hope to make an "excellent recovery", his trauma surgeon told the Associated Press on Friday.
The hospital noted that this will be their final update for the weekend.More news: Congressional Baseball Game carries on, one day after shooting
She says she's grateful to President Donald Trump, Vice President Mike Pence and all of Scalise's Capitol Hill colleagues "who have reached out to us during this most challenging time". Capitol Police officer David Bailey and Texas Rep. Roger Williams were also injured during the incident.
He had with him a piece of paper with doodles and the names of three politicians, according to a person familiar with the case. Hodgkinson, 66, of Belleville, was fatally shot in a firefight with police.
Lobbyist Matt Mika was shot multiple times and remains hospitalized.
Supporters of House Majority Whip Steve Scalise, R-La., hold signs before the Congressional baseball game, Thursday, June 15, 2017, in Washington. Hodgkinson has been identified as the suspect in the Wednesday, June 14, 2017, Washington D.C. shooting.
Scalise is only afforded a security detail because he is the third most powerful Republican in the lower chamber, though it's likely that Washington lawmakers will see a beefing up of their security.