The South Dakota lawmaker, 59, was on "an uncomplicated post-operative course," the U.S. Capitol physician said after visiting him Thursday afternoon. Johnson suffered a hemorrhage in his brain caused by a rare and sometimes fatal condition.
"He has been appropriately responsive to both word and touch. No further surgical intervention has been required," said the physician, Adm. John Eisold. He had said earlier, "The senator is recovering without complication."
Without more details, it is impossible to speculate on Johnson's prognosis, but CBS News medical correspondent Dr. Jon LaPook reports that, in general, patients with Johnson's condition can make a complete and total recovery.
Incoming Senate Majority Leader Harry Reid, D-Nev., said he had visited Johnson in the hospital Thursday morning and that he was confident of a full recovery. "To me he looked very good," Reid said.
Johnson was stricken as Democrats prepared to take fragile 51-49 control of the new Senate when it convenes in three weeks. Democrats seized control of both chambers of Congress from Republicans in November midterm elections.
If Johnson were to leave office, a replacement would be named by South Dakota's Republican governor, Mike Rounds. A Republican appointee would create a 50-50 tie and effectively allow the GOP to retain Senate control because Vice President Dick Cheney holds the tie-breaking vote.
One Republican told CBS News correspondent Gloria Borger, "This is not the way we want to win back the Senate."
Asked about whether Democratic control of the Senate might be jeopardized, Reid said, "There isn't a thing that's changed."
Johnson was rushed to the George Washington University Hospital at midday Wednesday after becoming disoriented and stammering during a conference call with reporters.
On Thursday, Johnson underwent an additional procedure to check for blood clots. The procedure is standard after surgery, said Julianne Fisher, Johnson's spokeswoman. Otherwise, she said, there were no new developments. "No news is good news," she said.
Dr. Keith Siller, medical director of New York University's Stroke Care Center, told CBS News that Johnson's recovery could range from weeks to months.
Johnson's condition, also known as AVM, or arteriovenous malformation, causes arteries and veins to grow abnormally large, become tangled and sometimes burst. The condition is often present from birth.
Johnson spokesman Noah Pinegar said the senator's diagnosis was a surprise. "No one was aware of it, including Tim," he said.
Reid, the Democrat who is to become majority leader when the new Senate convenes on Jan. 4, said, "We're all praying for a full recovery. We're confident that will be the case."
However, Reid declined to provide any details of Johnson's medical condition.
"The Republicans selected their committees yesterday. We've completed ours," Reid said. He said he was keeping incoming Senate Republican leader Mitch McConnell "totally advised" of developments.
Outgoing Senate Majority Leader Bill Frist visited the hospital Thursday.