But McClellan declined to say whether the president felt it should have been revealed earlier.
"I think that the vice president clearly explained the rationale behind that," McClellan said, avoiding a direct response to questions about whether the president felt the incident was publicly disclosed in a timely manner.
The accident happened on Saturday but was not publicly revealed until the next day.
The shooting victim, 78-year-old attorney Harry Whittington, remained in stable condition in a Texas hospital Thursday, two days after doctors said that a shotgun pellet entered his heart and he had what they called "a mild heart attack."
Hospital officials said Whittington was "doing wonderfully and superb" and they expected no further complications.
Cheney spoke publicly about the accident for the first time Wednesday in an interview with Fox News Channel. He said he did not see Whittington until just after he fired on a covey of quail and peppered him with bird shot in the face, neck and chest.
Cheney described it as "one of the worst days of my life" and rejected the notion that Whittington bears any responsibility. "I'm the guy who pulled the trigger and shot my friend," Cheney said.
"The president's satisfied with what the vice president said yesterday," McClellan said Thursday. "I think the American people are looking at this and saying, enough already," he added.
Check out the Texas Parks and Wildlife Department's report on Cheney shooting Whittington (.pdf).
Read and comment about coverage of this story in Public Eye.
Asked whether Mr. Bush encouraged Cheney to speak out, McClellan said, "Obviously there are internal discussions that we have, and I'll leave those internal."
But sources tell CBS News it was Mr. Bush's top political aide, Karl Rove, who pushed Cheney to speak publicly about the shooting.
Rove worried the vice president's silence on the issue was becoming a political problem, CBS News chief White House correspondent Jim Axelrod reports.
Cheney is in a "state of meltdown" over shooting his friend and the political fallout it has caused, a source close to Cheney said.