Doctors said her recovery will still be a very long process, and Tuesday will mark a crucial time to see how serious any brain swelling may be.
Giffords, a three-term Democrat, was gravely wounded in a shooting rampage in Tucson, Ariz., on Saturday that left six people dead, including a federal judge and 9-year-old Christina Green.
Reporters waited outside the Tucson home of Loughner's parents Tuesday afternoon, hoping the couple would speak for the first time since Saturday's tragedy.
The 22-year-old's mother and father released a statement:
"There are no words that can possibly express how we feel. We wish that there were so we could make you feel better. We don't understand why this happened. It may not make any difference but we wish that we could change the heinous events of Saturday. We care very deeply about the victims and their families. We are so very sorry for their loss."
Bill Hileman, the husband of the woman who took Green to the event, spoke out during the Tuesday hospital briefing.
"I hear her in her semi-conscious ramblings screaming out, 'Christina, Christina, let's get out of here, let's get out of here,'" Hileman said.
Authorities said the shooting was an assassination attempt. In the wake of the tragedy, federal security officials are planning to re-evaluate how Congress is protected while they are in their home districts. Currently lawmakers do not have any protection. At least two congressmen said they carry their registered guns more often.
The accused gunman, 22-year-old Jared Loughner, appeared with a shaved head in a Phoenix federal courtroom for the first time on Monday.
He showed no emotion and only briefly spoke when the judge asked if he understood that he could get life in prison or the death penalty. He replied, "Yes."
Loughner is being held without bail.
Fourteen people were injured in the attack, including Giffords.
President Barack Obama and first lady Michelle Obama are scheduled to arrive in Arizona on Wednesday for a memorial service. Funeral services for the shooting's youngest victim, 9-year-old Christina Taylor Green, will be on Thursday.
ABC News and the Associated Press contributed to this report.