Bremer insisted that the transfer of sovereignty to Iraqis must go ahead as scheduled despite the killing.
"They will not succeed. We must continue the political process leading to an interim government next month and to elections next year," Bremer said, whose American-run coalition is struggling to contain an insurgency in Sunni areas north and west of Baghdad, as well as an uprising in the Shiite heartland to the south.
In other developments:
At Saleem's memorial, his family and members of the Governing Council gathered inside the so-called Green Zone, which houses the coalition headquarters. Iraqi security forces in desert camouflage carried Saleem's coffin, which was draped with the Iraqi flag, from the hall after the service.
Bremer and other dignitaries, including new Governing Council chief Ghazi Mashal Ajil al-Yawer, kissed and shook hands with Saleem's grieving relatives.
"The enemies of freedom have taken him from his countrymen, from his family and from his friends, and although his loss is greatest for his family and for his country, those of us in the coalition shall miss him as well," Bremer said.
U.N. envoy Lakhdar Brahimi, who is consulting council members and other Iraqis about the makeup of the interim government that will take office after the power handover on June 30, was also at the ceremony.
"We're all working together in order to rebuild Iraq, which he sacrificed his life for," Brahimi said. "We express our cordial sorrow to his relatives and to the Iraqi people."
The killing was a major setback to American efforts to stabilize Iraq just six weeks before the handover of sovereignty.
In London, British Foreign Secretary Jack Straw said Tuesday that the situation in Iraq is tougher than Britain anticipated last year.
"It's palpable that the difficulties which we have faced have been more extensive than it was reasonable to assume nine months ago," Straw told British Broadcasting Corp. radio.
He said many of the security problems in Iraq had arisen following the attacks on the U.N. headquarters in Baghdad. The first on Aug. 19 killed 22 people, including top U.N. envoy Sergio Vieira de Mello.
Saleem, also known as Abdel-Zahraa Othman, was waiting in a Governing Council convoy at a U.S. checkpoint along a tree-lined street preparing to enter the Green Zone when the bomb was detonated. It apparently had been rigged with artillery shells and hidden inside a red Volkswagen.
Two other well-known council members, Adnan Pachachi and Ahmad Chalabi, reportedly escaped the blast because their cars had made it inside the compound — safely behind the concrete blast barriers, reports CBS News Correspondent Jim Acosta.
Iraqi officials said nine people, including the bomber, were killed and 14 Iraqis and an Egyptian were wounded in Monday's attack. Military spokesman Brig. Gen. Mark Kimmitt put the death toll at seven. Two U.S. soldiers were slightly wounded.
Saleem, a Shiite Muslim in his 60s, held the rotating presidency of the 25-member Governing Council for May. He was the second council member slain since their appointment last July; Aquila al-Hashimi was mortally wounded by gunmen in September.
Insurgents also have targeted police and army recruitment centers and other Iraqis perceived as owing their positions to the Americans.
The U.S. military said the car bombing was a suicide attack and Kimmitt said it had the "classic hallmarks" of Abu Musab al-Zarqawi, a Jordanian-born militant with links to al Qaeda.
However, a previously unknown group, the Arab Resistance Movement, claimed responsibility, saying in a Web site posting that two of its fighters carried out the attack on "the traitor and mercenary" Saleem.
Kimmitt said he did not know if the Arab Resistance Movement was "a cover for the Zarqawi network or if it's an actual organization."
Al-Zarqawi is believed responsible for many of the vehicle bombs in Iraq in recent months and for the beheading last week of U.S. civilian Nicholas Berg.
The Governing Council selected al-Yawer, a Sunni Muslim civil engineer from Mosul, to succeed Saleem. Al-Yawer will lead the council until the June 30 transfer of sovereignty to the new interim Iraqi government.