Scalise remains in critical condition following surgery, with the hospital treating him saying in a statement Wednesday night "he will require additional operations". The bullet caused massive bleeding, and damaged bones and internal organs.
Pence tweeted shortly after leaving the MedStar Washington Hospital Center on Thursday morning that he and his wife had thanked doctors and hospital staff.
Scalise was awake at the scene of the shooting - a baseball field in Virginia where congressmen were practicing for the Congressional Baseball Game - but by the time he arrived at the hospital he was in shock.
Thousands of people packed into Nationals Park Thursday night to watch the Republicans take on the Democrats in the 56th annual Congressional Baseball Charity Game.
Wednesday's attack was the first shooting of a member of Congress since January 2011, when Democratic Representative Gabby Giffords was seriously wounded in an assassination attempt at a gathering of constituents in Tucson, Arizona.
"We are deeply saddened by this tragedy", Mr. Trump said. Five people were shot and two others suffered secondary injuries.
The Capitol's usual partisan combativeness, though not absent, was less pointed than usual, and warm words were exchanged between some lawmakers with starkly opposing political philosophies.
"It's been much more hard than people even thought at that time", Trump said in a statement from the White House.
Also shot were a congressional staffer and members of the Capitol police force.
When the members of the Republican team were announced at Nationals Park, mention of Scalise's name drew a standing ovation from the areas designated for Republican, Democrat and nonpartisan fans alike.
"While the president would like to attend the game and show his support for all of these courageous public servants, he has been advised that there is not enough time to follow Secret Service protocols", the press secretary said.
Trump says, "Steve is our friend, he's a patriot and he's a true fighter".
Mr Trump also said at the White House that "Steve, in his own way, may have brought some unity to our long-divided country". Congressional aide Zack Barth, Tysons Food lobbyist Matt Mika and Capitol Police Officer David Bailey were also shot in the attack, but have since been released from the hospital.
Wednesday's shooting revived debate about gun rights in America.
The managers of the Democratic and Republican charity baseball teams said Thursday's game will go on as scheduled. Hodgkinson was eventually killed by Capitol police.
"The guy's down to a handgun, he dropped his rifle, they shoot him, I go over there, they put him in handcuffs", Conaway said, adding that if the shooter had "gotten inside the fence, where a bunch of guys were holed up in the dugout, it would have been like shooting fish in a barrel".