"We're not out of the woods yet," said Dr. G. Michael Lemole, section chief of neurosurgery at University Medical Center in Tucson.
"At this phase in the game, no change is good and we have no change," Lemole said at a briefing on the condition of Giffords and others wounded in the Saturday shooting rampage outside of a Tucson grocery store.
Giffords was shot in the head, with the bullet passing through the left side of her brain, back to front, during the attack, in which six people were killed and 14 wounded.
Among the dead were U.S. District Judge John Roll, presiding federal judge for Arizona, and Gabe Zimmerman, an aide to Giffords.
Lemole said CAT scans indicated there's been "no progression of ... swelling" in Giffords' brain and she can follow simple commands, NPR reported.
The doctor said "every day that goes by and we don't see an increase" in swelling is reason for cautious optimism.
Peter Rhee, the hospital's trauma chief, said eight patients remained hospitalized, two in intensive care.
In Washington, a moment of silence to honor the victims was observed by congressional staff members gathered on the Capitol steps, and by President Barack Obama, first lady Michelle Obama and White House officials on the South Lawn.
At the White House observation, the flag was at half-staff and a chime rang three times.
Obama signed a proclamation ordering flags at all public and military facilities to be flown at half-staff "as a mark of respect" for the victims.
The Obamas both canceled events scheduled for Tuesday. The president was to travel to Schenectady, N.Y., and the first lady canceled a public appearance with business leaders. Both events will be rescheduled.
The House of Representatives canceled a Wednesday vote on a bill to repeal healthcare reform legislation and instead will consider at least one resolution honoring Giffords and the other shooting victims, House leaders said. The House wasn't in session Monday and planned a pro-forma session Tuesday in which no votes or legislative business would be conducted.
Two house Democrats are drafting possible legislative responses to the Tucson incident, CQ Roll Call reported.
Rep. Carolyn McCarthy of New York is drafting a gun-control measure that would curb the availability of the type of high-capacity ammunition clip that Jared Loughner, the suspect in the Arizona attacks, allegedly attached to his pistol. Rep. Bob Brady of Pennsylvania has prepared legislation that would make it a federal crime to use language or symbols that could be considered as threatening or encouraging violence against congressional members.
In his first public statement since the shooting, Giffords' husband, astronaut Mark Kelly, thanked supporters and offered condolences to families of other victims, CNN reported Monday.
"Many of you have offered help. There is little that we can do but pray for those who are struggling," Kelly said in the statement. "If you are inspired to make a positive gesture, consider two organizations that Gabby has long valued and supported: Tucson's Community Food Bank and the American Red Cross."
Giffords, a three-term Arizona Democrat, was "doing what she loved most -- hearing from her constituents -- when this tragedy occurred," Kelly's statement said. "Serving southern Arizonans is her passion, and nothing makes her more proud than representing them in Congress."