However, the Prime Minister also waded into a domestic row as he accused MPs who voted against action last week of failing to "stand against the gassing of children".
Labour hit back, saying Mr Cameron was "demeaning his office by throwing around petty accusations at Ed Miliband".
The Prime Minister's comments are likely to infuriate those within his own backbenches who decided they could not back his strategy for intervention.
As he met Barack Obama and other world leaders at the G20 summit in Russia, the Conservative leader said of last week's shock defeat: "Everyone who voted has to live with the way that they voted.
"In my view, they (opponents) chose the easy and political path, not the right and the difficult path."
Mr Cameron also warned that a lack of action by the Americans would send an appalling signal to dictators across the world.
Syria is set to dominate the summit, which had been due to concentrate on economic matters.
The international response to the growing humanitarian crisis in Syria was originally not due to form part of the talks between the world's 20 largest economies.
However, amid growing pressure it emerged yesterday that it would be discussed, despite resistance from Russia.
Syria is also expected to dominate bi-lateral discussions between countries "on the margins" of the summit.
Mr Cameron has ruled out UK action in Syria following the Commons vote.
But he insisted yesterday the UK would not be marginalised, despite reports a Russian spokesman dismissed Britain as a "small island no-one pays attention to".
The UK would be one of the leaders involved in bringing forward talks of a peace process in Syria, Mr Cameron said.
On America, he added: "Having set a red line on the further big use of chemical weapons, I think it would be wrong if America was to step back, and having set that red line, to do nothing, I think that would send an appalling signal to President Assad, and also to dictators elsewhere.
"As I say, the House of Commons decision, as I interpret it, is that there should be no British involvement in that military action, and I respect that, but the world does need to respond strongly, and I won't stop making that argument."
Evidence the Assad regime was behind a gas attack in the Syrian capital was growing, he added.
He said: "I think the evidence is growing all the time, and we have just been looking at some samples taken from Damascus in the Porton Down laboratory in Britain which further shows the use of chemical weapons in that Damascus suburb.
"For some people there will never be enough evidence and for some in the debate in the House of Commons it wasn't about evidence, it wasn't about chemical weapons. It was about how they felt let down over Iraq, and it was a deep concern, which I completely understand, of not wanting to get further involved in the difficulties in Syria.
"My view is we have to look at chemical weapons as something different, something awful. Those pictures of children being gassed on our television screens are something the world must not turn away from."
In America, the full Senate is likely to begin voting on Wednesday, a Senate aide said. It will start with a vote on an anticipated legislative roadblock by Republicans, and then move on to a vote on the resolution to authorise the use of force.
Senate Majority Leader Harry Reid was "guardedly optimistic" the resolution would pass by the end of next week, the aide said.
President Barack Obama has cancelled a trip to California next week so that he can stay in Washington "to work on the Syrian resolution before Congress", according to a White House official.