David Cameron has been accused of allowing a secret rightwing agenda to flourish at the heart of the Conservative party, as fallout from the resignation of Liam Fox exposed its close links with a US network of lobbyists, climate change deniers and defence hawks.

In a sign that Fox's decision to fall on his sword will not mark the end of the furore engulfing the Tories, both Liberal Democrat and Labour politicians stepped up their demands for the prime minister to explain why several senior members of his cabinet were involved in an Anglo-American organisation apparently at odds with his party's environmental commitments and pledge to defend free healthcare.

At the heart of the complex web linking Fox and his friend Adam Werritty to a raft of businessmen, lobbyists and US neocons is the former defence secretary's defunct charity, Atlantic Bridge, which was set up with the purported aim of "strengthening the special relationship" but is now mired in controversy.

An Observer investigation reveals that many of those who sat on the Anglo-American charity's board and its executive council, or were employed on its staff, were lobbyists or lawyers with connections to the defence industry and energy interests. Others included powerful businessmen with defence investments and representatives of the gambling industry.

Fox's organisation, which was wound up last year following a critical Charity Commission report into its activities, formed a partnership with an organisation called the American Legislative Exchange Council. The powerful lobbying organisation, which receives funding from pharmaceutical, weapons and oil interests among others, is heavily funded by the Koch Charitable Foundation whose founder, Charles G Koch, is one of the most generous donors to the Tea Party movement in the US. In recent years, the Tea Party has become a potent populist force in American politics, associated with controversial stances on global warming.

Via a series of foundations, Koch and his brother, David, have also given millions of dollars to global warming sceptics, according to Greenpeace.

Labour said it wanted to know how, in 2006, when David Cameron travelled to Norway for his famous photo opportunity with huskies to promote his new-look party's "green" policies, his senior colleagues were cosying up to US groups that were profoundly sceptical about global warming.

Writing in the Observer, the shadow defence secretary, Jim Murphy, said the Tories still had many questions to answer and claimed that "while David Cameron's compassionate conservatism has been undermined by his actions at home, it could be further damaged by connections overseas".

Murphy writes: "With each passing day there have been fresh allegations of money and influence and it appears that much of the source was the Atlantic Bridge network and its US rightwing connections. We need to know just how far and how deep the links into US politics go. This crisis has discovered traces of a stealth neocon agenda. For many on the right, Atlanticism has become synonymous with a self-defeating, virulent Euroscepticism that is bad for Britain."

Fox resigned on Friday after admitting that he had allowed his friendship with Werritty, a lobbyist who portrayed himself as an adviser to the defence secretary, to blur his professional and personal interests. His resignation followed a drip-feed of revelations about the links between Werritty and businessmen and organisations with defence interests.

The revelations over Atlantic Bridge have triggered questions about the role played by Fox, chair of the charity's advisory council, and that of four of its UK members: William Hague, George Osborne, Chris Grayling and Michael Gove. As a UK charity, the organisation enjoyed tax breaks but had to comply with strict rules prohibiting it from promoting business interests.

The charity's political agenda, which it articulated in conferences devoted to issues such as liberalising the health sector and deregulating the energy markets, chimes with the thinking of many on the right of the Conservative party whom Cameron has been keen to check as he holds the Tories to the centre ground of British politics.

Lib Dem peer Lord Oakeshot said: "Dr Fox is a spider at the centre of a tangled neocon web. A dubious pattern is emerging of donations through front companies. We need to establish whether the British taxpayer was subsidising Fox and his frontbench colleagues. What steps did they take to ensure Atlantic Bridge didn't abuse its charitable status?"

Werritty, the group's UK director, was funded by a raft of powerful businessmen including Michael Hintze, one of the Tories biggest financial backers whose hedge fund, CQS, has investments in companies that have contracts with the Ministry of Defence; Poju Zabludowicz, chairman of the Britain Israel Communications and Research Centre, who chairs a US munitions company; and the Good Governance Group, a private security firm set up by a South African businessman, Andries Pienaar, who also has an investment firm, C5 Capital, focused on the defence sector.

The potentially explosive mix of big business interests and politicians that triggered Fox's demise is the subject of an investigation by the cabinet secretary, Sir Gus O'Donnell. Murphy said it was essential that the government then referred the wider issues to Sir Philip Mawer, the independent adviser on ministers' interests. "He should look at the issues in their entirety to establish precisely how this never happens again," Murphy said.

Questions are being asked over the role played by an organisation called the Sri Lankan Development Trust, whose headquarters were listed at the Good Governance Group. The trust paid for three of Fox's trips to Sri Lanka. In a statement the group said: "Our involvement with the Sri Lankan Development Trust was not done for profit or at the behest of any clients."

Arriving at the Ministry of Defence to take up his new role in charge of the department, Philip Hammond, the new defence secretary, said Fox had "done a great job".