One Republican also had a warning for the CIA. "Heads should roll, people should go to jail if it's true," Sen. Lindsey Graham, R-S.C., said.
But he appeared to be in a minority within his party.
Sen. Saxby Chambliss of Georgia, the top Republican on the Intelligence Committee, said he disagreed with Feinstein on the dispute with the CIA, without fully specifying how. "Right now we don't know what the facts are," he told reporters. "We're going to continue to deal with this internally."
A second committee Republican, Sen. Richard Burr of North Carolina, declined to comment, saying he had not yet read Feinstein's speech.
Sen. Mitch McConnell of Kentucky, the party's leader, declined to comment on the ongoing investigation into what happened.
At the White House, spokesman Jay Carney sidestepped most questions on the subject and reminded reporters, "We are talking about an investigation into activities that occurred under the previous administration" and which President Barack Obama ended soon after taking office.
Carney said Obama wants the report's findings to be declassified eventually.
There were suggestions that yet another investigation be established to look into Feinstein's charges and Brennan's rebuttal, a process that could add months if not years to a public accounting of detentions and interrogations that occurred a decade or more ago.
The activities at issue were approved by the George W. Bush administration and carried out by the CIA in the years after the terrorist attacks of Sept. 11, 2001.
Obama outlawed their use when he became president in January 2009. The committee began an investigation two months later, and the CIA provided access to documents totaling more than 6.2 million pages, Feinstein said.
The Senate committee staff wrote a 6,300-page report that the panel approved in December 2012, and the CIA provided a formal response six months later. Neither the full report nor a shorter summary has been released to the public.
In her speech, Feinstein accused the CIA of possible criminal activity in improperly searching the computer network set up for lawmakers investigating allegations that the agency used torture in terror investigations during the Bush administration.
In addition, more than 900 pages of documents the CIA initially made available to Senate aides were inexplicably withdrawn in the first few months of 2010, she said.
"I have asked for an apology and a recognition that this CIA search of computers used by its oversight committee was inappropriate. I have received neither," she said.
Associated Press writers Bradley Klapper, Donna Cassata, Stephen Braun and Julie Pace contributed to this report.
- Ex-MLB hurler winds up for Vermont governor...
- Origins of key Clinton emails from report are...
- Egypt says search for crashed EgyptAir plane...
- In Hiroshima, Obama honors 'silent cry' of...
- The last tackle powder puff football game...
- Donald Trump breaks with nation's only Latina...
- Rubio says he will support Trump, but not...
- NASA slowly inflating new room at space...
- Are Utahns tiring of Mitt Romney... 113
- Why the University of Miami plans to... 45
- In Hiroshima, Obama honors 'silent cry'... 25
- Donald Trump breaks with nation's only... 18
- Trump: Those in US illegally treated... 18
- Trump tells California 'there is no... 17
- On cutting-edge voter data, Trump... 14
- Origins of key Clinton emails from... 13