A doctor treating the Arizona congresswoman said he was "very optimistic" about her chances to recover after she had been shot "through and through" the head.
Dr. Peter Rhee, trauma director of the University of Arizona Medical Center, said Giffords, 40, was in critical condition following surgery but was able to "follow commands" in the hospital.
Speaking from the White House, Obama said it was not known what provoked the "unspeakable act" but said, "We are going to get to the bottom of this and we will pull through this."
A total of 18 people were shot, said Pima County Sheriff's Deputy Chief Rich Kastigar.
He identified the suspect as a Jared Lee Loughner, 22, from Arizona and said he was armed with a pistol that had "an extended magazine."
Among the dead is U.S. District Judge John Roll, the chief federal judge in Arizona, and a nine-year-old girl.
Obama said he had dispatched FBI Director Robert Mueller to Tucson to help lead the investigation.
The U.S. Capitol Police said it was working with federal, state and local law enforcement agencies and that it has contacted House members "advising them to take reasonable and prudent precautions regarding their personal safety and security."
Giffords was holding an event outside a Safeway grocery store when a man ran up and started shooting, witnesses said.
The Democrat narrowly defeated tea party-backed Republican Jesse Kelly to win a third House term in November.
The election capped a year of heated political rhetoric directed at Giffords and other Democrats who supported the new health care law.
In March, Giffords' congressional office in Tucson was vandalized after she voted in favor of the bill. Sarah Palin included Giffords on a list of Democratic candidates in the crosshairs for defeat in the midterm elections because of their support for the overhaul.
When asked by the New York Post whether his daughter had any enemies, Spencer Giffords, 75, tearfully replied: "The whole tea party."
Palin, whose map of targested Democrats was taken down soon after the shooting, posted condolences on her Facebook page, saying, "On behalf of Todd and my family, we all pray for the victims and their families, and for peace and justice."
Statements of shock and sympathy poured forth at the news.
Speaker of the House John Boehner called it "a sad day for our country" and said he was "horrified" by the shooting. "An attack on one who serves is an attack on all who serve," he said.
House Demoratic Leader Nancy Pelosi said, "This terrible act of violence is a national tragedy, and today is a very sad day for our country."
Gov. Jan Brewer said her state is "shocked" and "saddened" and said the attack "certainly doesn't show favor on the state of Arizona."
Democratic strategist Donna Brazile reacted to the shooting with this message on Twitter: "The increasing levels of incivility, intolerance and a political climate that favors those who threatens, bully others must end. I am so sad."
Giffords has served on the House Committee on Foreign Affairs and the House Committee on Armed Services, according to her election website.
Giffords previously served in the Arizona legislature and State Senate and was CEO of El Campo Tire, her family's tire and automotive business.
The election website describes Giffords as "a longtime gun owner and strong supporter of the Second Amendment."
In 2007, Giffords married astronaut Mark Kelly, who rushed to Tucson to be at his wife's side. Kelly's twin brother, Scott, is currently in orbit, serving as commander of the International Space Station.