"Mr Saleem died working to build a free, democratic and prosperous Iraq," said presidential spokesman Scott McClellan. "The Iraqi people will continue his work and see to it that such a vision becomes a reality.
"The enemies of a free, democratic and peaceful Iraq (search) will not prevail," McClellan said.
Saleem, also known as Abdel-Zahraa Othman, was killed in a suicide car bombing near a checkpoint outside the coalition headquarters in central Baghdad. He was the second and highest-ranking member of the U.S.-appointed council to be assassinated.
A State Department spokesman, Adam Ereli, said it was not clear whether he had been targeted for assassination. Nor can the Bush administration identify any group as being responsible.
In any case, Ereli said, "it's a loss for the Iraqi people."
"Democracy and freedom are taking root and these cowardly, terrorist acts only make us more determined to finish our work," McClellan said. "We strongly condemn this attack." He said Americans' thoughts and prayers were with Saleem's family.
Sen. Dick Lugar, R-Ind., chairman of the Foreign Relations Committee, called the assassination "a very tragic act."
"Whether he was selected specifically because he was the president for this month or because he was fairly moderate, we do not know," Lugar said on C-SPAN.
"But I think it illustrates ... these Iraqis are going to be in the firing line. They are going to be targets. This is very dangerous business ... as we're negotiating both aspects, democracy and security."
"Today's tragic incident illustrates how perilous this transition will be," Lugar said.