President Trump described the attack as a "very, very brutal assault".
"We may have our differences, but we do well in times like these to remember that everyone who serves in our nation's capital is here because, above all, they love our country," he added.
The president visited Mr Scalise - who was shot in the left hip and is in a critical condition after surgery - at the MedStar Washington Hospital Center on Wednesday evening, describing him as a "patriot and a fighter".
The hospital said Mr Scalise had suffered franctured bones, internal organ injuries and severe bleeding and would require further operations.
What we know about suspect
James T Hodgkinson is a 66-year-old man from Belleville, Illinois
His wife told ABC News he moved to Virginia two months ago
His Facebook account shows anti-Republican and anti-Trump rhetoric
He volunteered on the campaign of former Democratic presidential candidate Bernie Sanders
He had a string of criminal convictions, including traffic violations and allegations of violence
The lawmakers were practising for the annual congressional ballgame that organisers vowed would go ahead as planned on Thursday at Nationals Park, home of Major League Baseball team the Washington Nationals.
Alexandria police said the FBI was taking over the investigation into the shooting, which began shortly after 07:00 (11:00 GMT) on Wednesday morning at Eugene Simpson Stadium Park.
As well as Mr Scalise, two police officers, Krystal Griner and David Bailey, lobbyist Matt Mika and Zack Barth, a congressional aide for Texas lawmaker Roger Williams, were injured.
Mr Barth, a former staffer for two Republican congressmen, was shot in the chest, according to witness Arizona Senator Jeff Flake.
Mr Flake told reporters he was at bat during the shooting and the victims were "sitting ducks".
Senator Mike Lee, a Utah Republican, said someone at the baseball field had used a belt as a tourniquet on Mr Scalise, who is the number three House Republican.
Kentucky Senator Rand Paul told Fox News: "I do believe without the Capitol Hill police it would have been a massacre.
"We had no defence at all. The field was basically a killing field."
He said they were lucky that Mr Scalise was present because he has a security detail owing to his congressional seniority.
Mo Brooks, an Alabama Republican, told CNN he was on third base during the baseball game when he saw Mr Scalise, who was on second base, shot.
He said Mr Scalise had a bullet hole in his leg, but was saying: "I'm OK, I'm OK."
Mr Brooks said everyone on the field scattered as 50 to 100 shots were fired.
He said a gunfight had ensued between the shooter and the congressmen's police security detail, who were armed with pistols.
Mr Brooks, who took refuge behind a batting cage, said the gunman was armed with a rifle and was "blasting away" from behind the dugout.
Mr Brooks said congressmen Chuck Fleischmann, of Tennessee, Ron DeSantis, of Florida, and Jeff Duncan, of South Carolina, were among 15-25 other lawmakers at the game.
Asked by a CNN host if the attack was deliberate, Mr Brooks replied: "Well, it sure as heck wasn't an accident!
"He was going after elected officials, congressmen."
Mr Brooks continued: "The only weapon I had was a baseball bat and that's not the kind of fight you want to engage in."
Ohio congressman Brad Wenstrup, who served in Iraq as a combat surgeon from 2005-06, provided medical care at the scene.
He said the attack had lasted for at least 10 minutes, adding: "He had a lot of ammo."
Alexandria Police Chief Michael Brown said his officers had responded and engaged the shooter within three minutes.
The last member of Congress to be targeted by a gunman was Democrat Gabrielle Giffords, who was shot in the head as she met constituents in Tucson, Arizona, in January 2011. She survived, but six others died in that incident.
Ms Giffords said in a statement: "This shooting is an attack on all who serve and on all who participate in our democracy."
The top Republican in the House of Representatives, Speaker Paul Ryan, was applauded on the chamber floor as he said: "We are united... an attack on one of us is an attack on all of us."