Critics had called for Rumsfeld to quit after the Iraqi prisoner abuse scandal.
But analysts, editors and ordinary Arabs were united in their condemnation of Bush on Monday when he said the nation owed the defence secretary a "debt of gratitude".
"After the torture and vile acts by the American army, President Bush goes out and congratulates Rumsfeld. It's just incredible. I am in total shock," said the editor of the influential Algerian national daily al-Watan.
"Bush's praise for Rumsfeld will discredit the United States ... and further damage its reputation, which is already at a historic low in the Arab world."
Analysts have said the damage from images seen worldwide of soldiers abusing naked Iraqi prisoners in Abu Ghraib prison would be indelible, incalculable and a gift to al-Qaida leader Usama bin Ladin.
What people saw, they said, was the true image of the occupation: humiliation of an occupied people, contempt for Islam, sadism and racism.
"Mr Rumsfeld would have certainly lost his job if the prisoners were American"
Dubai-based political analyst
"After Mr Bush's decision to keep Rumsfeld, all their apologies seem like lip service," said Dubai-based political analyst Jawad al-Anani.
"Mr Rumsfeld would have certainly lost his job if the prisoners were American."
Symbol of Iraq war
University of Algiers professor Mahmoud Belhimeur agreed.
"I cannot believe the United States reacts the way an authoritarian regime would. Bush should have done the honourable thing and fired Rumsfeld," he said.
But Michael Cox, professor of international relations at the London School of Economics, said the repercussions of firing the defence secretary would have been very significant.
"This has been Rumsfeld's war, and I suppose the political symbolism of trying to get rid of Rumsfeld would be huge."
Cox said he could not entirely rule out that Rumsfeld could go, if US public opinion turned. But he added it would seem out of character for the defence secretary to go quietly.