The chief secretary to the Treasury, Danny Alexander, has said the EU veto does not threaten the coalition, defending the Conservatives' and Liberal Democrats' right to disagree over David Cameron's decision to block a new EU agreement.

Amid fears that the rift between the two parties over Cameron's veto could be fatal, the senior Lib Dem cabinet minister said Nick Clegg was entitled to speak out but expressed sympathy for the prime minister's actions, which were described by Clegg as a "bitter disappointment".

"Of course it's right that Nick Clegg is able to speak out as he has … [but] David Cameron had a very difficult hand to play," Alexander told BBC Radio 4's Today programme on Monday. He said the prime minister faced "real intransigence from France and Germany and a clear need to bring something back to the House of Commons to show his own supporters".

Clegg had lashed out at the decision, warning that it could lead to the UK being "isolated and marginalised within the European Union". But Alexander took a conciliatory approach, saying: "There is no threat to the coalition. The coalition was formed to deal with the enormous economic problems that we inherited as a country. That task is the central task of this government."

While emphasising the commitment to Europe as one of the "fundamental beliefs" of the Liberal Democrats, he said the party could point to gains within the coalition, including additional funding for the pupil premium and the commitment to increase the income tax personal allowance to £10,000.

Alexander said agreement between all 27 EU member states would have been the "best outcome for Britain" but only if there were safeguards for the financial services industry.

Cameron has indicated that he will not allow the 26 EU countries who agreed on greater financial integration last week to use the union's institutions to administer the new pact, a stance described by Clegg as "ludicrous", but Alexander refused to be drawn on the issue. "We don't know yet how they would use these institutions or even if they want to use the institutions," he said.

The prime minister is due to address MPs in the House of Commons at 3.30pm on Monday to explain his decision to veto the EU treaty changes. He is likely to receive a warm welcome from Eurosceptic Conservative MPs who, buoyed by the decision, want Westminster to reclaim powers from Brussels in return for future co-operation. But there is growing anger among the ranks of the Tories' coalition partners and the Lib Dems have pleaded with the prime minister to avoid triumphalism in his statement.

Clegg said those welcoming the outcome of the summit were "spectacularly misguided". "There's nothing bulldog about Britain hovering somewhere in the mid-Atlantic, not standing tall in Europe, not being taken seriously in Washington," he said.

Several high-profile Lib Dem figures, including the former leader Paddy Ashdown and the party president, Tim Farron, joined Clegg in a wide-ranging attack on Cameron's move over the weekend.

A Populus poll for the Times showed support for Cameron's position. The survey showed 57% felt he was right to use the veto while only 14% disagreed with the move. While 56% of voters felt the UK would lose influence in the EU, only 24% felt that it would weaken the prospect of economic recovery in Britain.

By Monday, senior Lib Dems went on air to stress that the coalition was not at risk as a result of last week's events in Brussels. But they made clear that the party would "face down" Tory backbench Eurosceptics.

Simon Hughes, the Lib Dem deputy leader, made clear his party would resist Eurosceptic Tory backbenchers to ensure that Britain retains a "constructive" approach to Europe: "There is no risk to the coalition. We came to an agreement in the national interest. Liberal Democrats and Tories – not natural allies – came to an agreement because we knew the country needed us to sort out the economy. That's why we did so.

"We are committed to that and it's a five-year plan, but it says very explicitly that we will be constructive about Europe and that means we will have to face down the people from the Tory backbenches who believe that being constructive about Europe is the wrong thing to do and we have to show that we are going to be a secure and positive player in the European Union. If not, we are at risk and the recovery of the economies of Europe is at risk."

Hughes echoed Clegg's concern that Britain risked becoming a "pygmy" on the world stage: "That's the danger … we are one country out of 27, there's a market of 500 million people, we're 62 million people. The reason we are so important is because we are part of the common market and if people think we will do better on the world stage alone, they are absolutely deluded."

He also questioned what the UK had achieved by vetoing the treaty and warned it could undermine Europe's economic recovery.

"The negotiation was done by David Cameron as prime minister. We don't have any argument with that, but the question is, what is the outcome? The outcome is that instead of the European Union countries looking as if they're going together to sort out the eurozone crisis – that was the major issue for the summit: were we going to sort out the eurozone crisis so we can deal with our economy, so we could get the economy going again, get growth, get jobs – it looks as if, from the outside, the UK is not part of that and we didn't get any prize to come back to give to the City, which apparently was the reason why we needed to take that position."

Labour's former foreign secretary, David Miliband, said Cameron had achieved nothing by his use of the veto, and warned that Britain now faced a "very dangerous moment".

"This is the first veto in history not to stop something. The plans are going right ahead. It was a phantom veto against a phantom threat," Miliband told Today.

"David Cameron didn't actually stop anything because the other 26 are going on and the provisions of the treaty would not have weakened our rights and freedoms one iota."

He added: "Fifty-six years ago Anthony Eden walked away from the founding of the European Union and we paid the price for 20 years. This has been an Anthony Eden moment for David Cameron," he said.

"If you look back to what the prime minister was warning of just two months ago – namely the danger of a caucus of 17 countries bossing the rest of the European Union – he has engineered a situation where it is 26 countries together and Britain on its own without a say for the first time since we joined the European Union."

David Laws, the Lib Dem former chief secretary to the Treasury, told BBC Breakfast that the difference between his party and the Conservatives was about the "tactical position" taken.

"The difference really between the parties isn't one about what the objectives of these talks were from the UK perspective, it is a disagreement about the tactical position that we got ourselves into as a consequence of not having other allies in the European Union who we can rely upon.

"The exercise of the veto has potentially put us in a position where we are one against 26, so we risk having the rules for the City and business set by other EU nations without our participation.

"That is why it's not just the Lib Dems, but it's many people in the City and business who are worried about whether we have protected our national interest, which we both agree must come first."

Meanwhile Boris Johnson, the London mayor, has insisted London will not suffer as a result of the decision to block the EU agreement as he backed the prime minister's move in Brussels.

The Conservative mayor also suggested that the events of last week were being viewed as "more dramatic than they really are" and warned that it would be "perverse" if European partners tried to put up trade barriers as a result.

During a speech at the South Bank centre Johnson appeared relaxed about Britain's position in Europe and said that he doesn't expect fellow members of the European union to "punish" Britain for its stance.

In a jokey aside to plans for greater fiscal union, which he publicly called "FU", Johnson insisted that Britain remained a "valued and paid-up member of the EU".

Highlighting his goals for the capital, the London mayor told a London audience: "We are not just a valued and paid up member of the EU – whether we are in the fiscal union, or FU, or whether we are in the second tier of the fiscal union, or FU2 - we not only have unrivalled links to America, we are the world capital of the Bric, with more Chinese students studying in London than any foreign city..."