The incident occurred Saturday but was not made public for nearly 24 hours – and then by a private citizen.
The shooting victim, 78-year-old attorney Harry Whittington, was recovering in stable condition Monday, a hospital official said.
Whittington "rested well last night," said Peter Banko, hospital administrator at Christus Spohn Hospital Memorial. The hospital listed Whittington's condition as "very stable," he said.
Banko said he did not know when Whittington would be released.
White House spokesman Scott McClellan was peppered with questions Monday about why it took so long for the incident to be disclosed. McClellan said the first priority was making sure Whittington was getting the medical care he needed, CBS News White House correspondent Mark Knoller reports.
Cheney apparently did not see Whittington, and the vice president accidentally hit him in the face, neck and chest with bird shot while the two were hunting quail.
The vice president is a lifelong hunter and ranch owner Katharine Armstrong said both he and Whittington were wearing orange safety vests, CBS News correspondent Kelly Cobiella reports.
White House chief of staff Andy Card told President Bush Saturday night about Cheney's involvement in the shotgun accident.
McClellan said he first learned Saturday night that someone in the Cheney hunting party was involved, but he didn't know that Cheney was the shooter until the next morning, the spokesman said.
McClellan said when he learned, around 6 a.m. Sunday, he urged the vice president's office to get the information out "as quickly as possible."
Kenedy County Texas Sheriffs Lt. Juan Guzman said deputies first learned of the shooting when an ambulance was called.
But the Secret Service told a different story, saying agents had informed the local sheriff of the shooting about an hour after it happened and that the vice president had been interviewed about the accident by local authorities on Sunday morning, CBS News White House correspondent Bill Plante reports.