The UK will "lead the world" in getting humanitarian aid to refugees in Syria, David Cameron has vowed.
The prime minister repeated his assurance that there will not be another vote in Parliament on military action after MPs rejected it last week.
But he said the Commons defeat does not mean he can "do nothing" to help those affected by the two-year civil war.
London Mayor Boris Johnson has said MPs should reconsider military action if the US opts to launch strikes.
But Mr Cameron appeared to rule out such an option, saying: "I think Parliament spoke very clearly and it's important to respect the view of Parliament, so I am not planning to return to Parliament to ask again about British military action."
He added: "Now, that does not mean we do nothing on Syria. We are already the second largest aid donor in delivering the humanitarian aid that is so needed, both in Syria and in the neighbouring countries like Jordan and Turkey.
"We'll go on doing that. We will help lead the world in that effort and make further efforts at the G20 to make sure that vital aid gets through."
Mr Cameron, who spoke during a school visit in Birmingham, said he would use the G20 meeting of world leaders in Russia later this week to press for help for the refugees as well as progress on diplomatic efforts to secure a stalled peace conference.
The United Nations has warned that one-third of Syria's population have been forced from their homes by the conflict, with another 4.25 million displaced within the country.
Britain's £348m commitment to help those affected by the Syrian conflict had saved "tens of thousands of lives", said Mr Cameron, adding: "It is difficult to get some of the aid into Syria itself but we must continue to try to crack these problems.
"Britain, as ever, is a world leader in helping those who need help and the people of Syria are right up there at the front of that right now."
Foreign Secretary William Hague earlier told MPs Britain "will do more" to help the relief effort.
US President Barack Obama is expected to use the G20 meeting in St Petersburg to rally other world leaders in support of US action, although his efforts are unlikely to sway G20 host host Vladimir Putin, a long-term backer of President Assad.
Mr Obama has indicated that he is "confident" of getting approval from Congress for a US military response to chemical weapons attacks on a rebel-held suburb of Damascus on 21 August.
Syria is not on the formal agenda for the G20 summit, but Downing Street made clear that Mr Cameron expects to take part in intensive discussions during the gathering.
Shadow foreign secretary Douglas Alexander, for Labour, said the UK government should have pushed for Syria to be part of the G20's formal agenda, rather than being discussed "on the margins".
But Foreign Secretary William Hague said the venue for discussions was not the issue, but how "we bring about a transitional government in Syria, formed from government and opposition by mutual consent".
He also told MPs Britain was prepared to talk to Iran about Syria, but cautioned MPs about the possibility of success, given that country's previous record on the issue.
"The test for Iran is whether it is prepared to play a constructive role, because Iran has, from all the evidence presented, been actively supporting the Syrian regime, including in the killing of so many innocent people in Syria. It hasn't played a constructive role so far but we are prepared to talk to it."
In other developments on Tuesday Israel carried out a joint missile test with the US in the Mediterranean, as Congress prepared for its first public hearings on whether the US should launch military action.
London Mayor Boris Johnson says he believed Congress would back President Obama in next week's vote and the British Parliament should "think again" about UK involvement in military action in Syria.
Asked what could be achieved by bombing Syria, Mr Johnson said: "It will show that in the end, when a tyrant decides to use unlawful weapons, gas, to murder innocent civilians, we will not stand idly by."
He told London radio station LBC 97.3: "I have been very hostile and sceptical about British involvement in Syria - we all remember what happened in Iraq. But if you see someone gassing innocent civilians, that is a chilling, chilling memory for people in Europe of what happened in the 1930s and 1940s and of the cost of doing nothing."
Mr Johnson added: "What I have said is if it can be proved that it really was the Assad regime that was responsible, and there's more proof, there's more evidence coming in, and if the Americans - who after all are the only real power that has the men and the material to deliver any retribution - can come up with a plan that is coherent, that is limited, that is punitive in some way but does not actually engage the West in some long-term commitment in a Syrian conflict, then I think there is a case for Britain looking again at whether or not we should participate."
If President Obama secured the backing of US politicians, Mr Johnson said "in those circumstances I think it would be appropriate for us to look at what new evidence there is, look at what the Americans are actually proposing, see whether it's coherent, see whether it's limited, see whether it's commensurate with what has happened, see whether it will deliver a plausible outcome".
He added: "I think it would be reasonable in those circumstances for Parliament to think again."