David Cameron's defeat over UK involvement in military action against Syria will lead to "national soul-searching" about the country's role in the world, Chancellor George Osborne has admitted.

The Prime Minister was forced to rule out the use of British forces in any action against Bashar Assad's regime after a humiliating vote in the Commons which saw 30 Tory rebels as well as nine Liberal Democrats side with Labour to defeat the Government.

Mr Osborne acknowledged that the inability to commit British forces to any American-led operation against Assad over his use of chemical weapons would damage the special relationship between Westminster and Washington.

He told BBC Radio 4's Today programme: "I think there will be a national soul-searching about our role in the world and whether Britain wants to play a big part in upholding the international system, be that big, open and trading nation that I like us to be, or whether we turn our back on that. I understand the deep scepticism that many of my colleagues in Parliament, many members of the public, have about British military involvement in Syria. I hope this doesn't become a moment where we turn our back on all of the world's problems."

The Chancellor said that in discussions with the White House since the shock vote in the Commons on Thursday night there had been a "lot of understanding" about the Government's decision to rule out military support. But Mr Osborne added: "Obviously it would have been better from the point of view of the special relationship if we were able to take part in any military action, should that military action take place, alongside the Americans. But we are not going to because we are going to respect the will of Parliament."

The Chancellor insisted there had been "hyperbole" over the depth of any damage to the relationship with the US.

Former Liberal Democrat leader Lord Ashdown said the defeat in the Commons had left the UK a "hugely diminished country". The former special forces soldier added on Twitter: "In 50 years trying to serve my country I have never felt so depressed/ashamed. Britain's answer to the Syrian horrors? none of our business!"

Ed Miliband urged the Government not to "wash its hands" of Syria. The Labour leader said David Cameron must "find other ways" to put pressure on President Assad. He said: "There are other routes than military means to actually help the people of Syria. I don't think the Government should wash its hands of this issue.

"I think all of the focus of the Prime Minister and the Government in the coming days needs to be working with our allies to find other ways to press President Assad, to take action with our allies to put the diplomatic, political and other pressure that needs to be put on the Government there. We need the peace talks to get going. So there are other things the Government should be doing."

The Commons motion backing the use of force "if necessary" in response to last week's deadly chemical weapons attack was rejected by 272 votes to 285, majority 13.