Blair announced Thursday he will resign June 27 as prime minister and leader of Britain's Labour Party.
President Bush was effusive about Blair, a crucial ally in the Iraq war, calling him a man who kept his word: "When Tony Blair tells you something, as we say in Texas, you can take it to the bank."
Bush also had kind words for Blair's anticipated replacement, Treasury chief Gordon Brown, saying he "found him to be an easy-to-talk-to, good thinker."
CBS News chief Washington correspondent Bob Schieffer said the trans-Atlantic relationship won't likely change much with Blair's departure.
"Nations do not necessarily have friends, they have interests. Tony Blair did what he thought was in Great Britain's interest," said Schieffer. "They will always be our closest ally, no matter who the prime minister there is or who the president of the United States is."
Former U.S. President Bill Clinton said: "Blair revitalized his party, modernized his country's economy and its approach to social problems, took the lead on global issues from climate change to debt relief to doubling aid to Africa, to the quest for peace in Northern Ireland and Kosovo, and started the global Third Way political movement."
"I am glad he was there and grateful for our friendship," Clinton said in a statement.
"Tony Blair has taken Britain from the fringes to the mainstream of the European Union," the president of the European Commission, Jose Manuel Barroso, said. Blair "leaves an impressive legacy including his commitment to enlargement, energy policy, his promotion of action against climate change, and for fighting poverty in Africa."
The local party faithful may still adore him, but the truth is much of Britain has fallen out of love with its prime minister, reports CBS News foreign correspondent Elizabeth Palmer.
His career was blighted, says Matthew Parris, by one key blunder.
"Iraq, Iraq, Iraq," Parris, a writer for The Times, told Palmer. "It must haunt him through his dreams now, and it will for the rest of his life."
And CBS News Foreign Affairs Analyst Pamela Falk notes, "Although the Anglo-American relationship will remain strong, Blair's likely successor, Gordon Brown, has strong ties with Democratic Party leaders in the U.S. and his foreign policy is likely to diverge from U.S. policy, particularly in Iraq, and particularly because of the criticism within the U.K. of the war."
On Friday, Blair's office said he had formally endorsed Brown to be prime minister.
"I'm absolutely delighted to give my full support to Gordon as the next leader of the Labour Party and Prime Minister and to endorse him fully," Blair said.
Swedish Prime Minister Fredrik Reinfeldt called Blair a "jovial, easygoing person" and a "fantastic speaker."
Staying in office for 10 years was a major accomplishment, Reinfeldt said. "And to do it in a country with the British press is an accomplishment in its own right," he said.