When Hagel tried to explain his position, McCain shot back, "Will you please answer the question?"
Hagel confirmed he thought what he called the "war of choice" in Iraq was the "most disastrous since Vietnam."
Hagel told McCain, "I would defer to the judgment of history" on the surge.
McCain insisted history already had judged the wisdom of the surge, and Hagel was wrong.
McCain tried to get Hagel to say whether he supported giving U.S. weapons to the Syrian opposition and establishing a no-fly zone.
Hagel said the United States was looking at those options, but McCain said 60,000 people already have died in the Syrian civil war -- "How many more would have to die before you would support [those actions?]" McCain asked.
McCain said what he considered Hagel's refusal to answer yes or no to his questions would influence whether he would vote to confirm Hagel, leaving little doubt that at this time the answer was negative.
McCain, the Republican presidential nominee in 2008, said Republicans had concerns about Hagel's "professional judgment."
Cruz said Hagel had not responded fully to requests for copies of his speeches and disclosure of his finances, particularly speaking fees, and whether some money came from foreign services.
Hagel started to respond, but Cruz cut him off, saying his time was limited.
Cruz then played two audio excerpts from an interview with al-Jazeera, in which Cruz claimed Hagel had agreed that Israel had committed war crimes and the United States was the world's nuclear bully.
Hagel responded by saying that was not what he had said in the interview.
The controversial tactic drew a rebuke from the committee chairman, Sen. Carl Levin, D-Mich., who said he had asked Cruz to supply transcripts of the interviews, but Cruz had chosen not to do that. Levin ordered that the transcripts be produced, and that Hagel be given a fair chance to answer them.
Levin later gave Hagel time to answer accusations about not supplying the panel with his speeches or finances.
"As far as I know, we responded to all requests [for copies of speeches or finances], or in the process of responding," Hagel said. "Some of these requests didn't come in till yesterday.
Hagel said all of his paid speeches were extemporaneous.
"I have complied with every ethical request" from the panel, he said.
Before Hagel's opening statement, the ranking member of the Senate Armed Services Committee, Sen. Jim Inhofe, R-Okla., indicated he would not support Hagel as defense secretary.
Inhofe said he told Hagel earlier "after a long and careful review, we're too philosophically opposed, and therefore I would not be supporting his nomination."
Inhofe said Hagel's record "is troubling and out of the mainstream," and his record is one of "appeasing our enemies and shunning our friends."
Hagel was introduced to the panel by two former chairmen of the committee, former Sen. Sam Nunn, D-Ga., and former Sen. John Warner, R-Va. Nunn cited Hagel's service in Vietnam and his two purple hearts, and strongly endorsed him.
"War for Chuck Hagel is not an abstraction," he said.
Warner said he would only say "a few words from the heart," adding he had read Hagel's statement to the committee, which he said outlines Hagel's precise goals. Warner said Hagel "took the point" as an infantry sergeant in Vietnam, and as defense secretary, "Chuck Hagel will do it again."
If confirmed, Hagel would be the first Vietnam veteran to head the Pentagon.
As a senator, Hagel broke with many in his party on the Foreign Relations Committee to criticize the management of the Iraq war after initially supporting the U.S.-led invasion.
Hagel will need 51 votes to win confirmation, unless Republicans threaten to filibuster. That could trigger a rule requiring 60 votes.
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