The "tragedy" of the Middle East crisis can be a chance to find a lasting peace for the whole region, Tony Blair has said, after talks with George Bush.
The two leaders have been criticised for their Middle East stance
The UK prime minister told reporters at the White House a new UN framework was needed to get the fighting stopped in Lebanon "urgently".
But he said the causes of the immediate conflict showed there had to be a lasting Israeli-Palestinian resolution.
Terrorists could not be allowed to derail hopes for democracy, he said.
"We really will never understand how we deal with this situation unless we understand there is a big picture out in the Middle East," he said.
That was "about reactionary and terrorist groups trying to stop what the vast majority of people in the Middle East want" - peace, democracy and human rights, he argued.
Mr Blair said he wanted to see Lebanon's government taking control of the whole country, for it to be the democracy people wanted and for there to be a resolution to the Israeli-Palestinian issue.
"If we are able, out of what has been a catastrophe for many of the people in the region, to do such a thing, then we would have turned a situation of tragedy into one of opportunity, and we intend to do that," he said.
Mr Blair and Mr Bush talked for just over an hour.
The prime minister is on his way to address the annual conference of Rupert Murdoch's News International media group in California.
He is facing mounting pressure from Labour MPs and others for him to join calls for an immediate ceasefire in Lebanon.
But he and the president stopped short of that call, saying that moves towards stopping the violence had to be brought forward on an "urgent basis".
Mr Bush said it was tempting to urge quick solutions to get the crisis off the television screens. In fact it had to be tackled properly for the long term, he argued.
And Mr Blair said: "We feel deeply for people in Lebanon and people in Israel who are the innocent casualties of this conflict, of course we do.
"And we want it to stop, and we want it to stop now.
"What we are putting forward today is actually a practical plan that would lead to a UN resolution...[to] put in place the conditions for it to stop."
Iran, Syria warned
The two leaders said they wanted a stabilisation force to be sent to the region.
A meeting at the UN to discuss the plan was being brought forward to Monday.
Mr Bush attacked Iran and Syria for sponsoring Hezbollah, while Mr Blair said the two countries faced a stark choice to either "come and participate as proper and responsible members of the international community or they will face the risk of increasing confrontation".
He warned that the Lebanon crisis would not mean any "sidetracking" away from ensuring that Iran complies with rules about its nuclear programme.
The White House talks came as demonstrators went to Downing Street to protest against UK policy on Lebanon.
Organisers said there were 1,500 to 2,000 people at the demonstration.
They handed over an open letter signed by 8,000 people urging Mr Blair to call for an immediate ceasefire.
A national demonstration is planned for 5 August in Hyde Park, by which time the Stop The War coalition hopes 50,000 people will have signed the letter.
Among the protesters was ex-Labour MP Tony Benn, who said: "We represent the overwhelming majority of people in Britain - two thirds - and we represent every country in the world that wants a ceasefire, except Israel, Bush and Blair."
Liberal Democrat leader Sir Menzies Campbell said he was disappointed with Mr Blair's comments.
"It seems as if the prime minister is impervious to the views of the majority in this country and of the wider world," he said.
"Every day of grace afforded to Israel and Hezbollah will result in more casualties and more destruction."