A decision by Mr Trump to pull the US out of the Paris Agreement - the world's first comprehensive deal on tackling climate change - and seek renegotiated terms that were "fair" to America has drawn widespread international condemnation.
Exclusive: Theresa May and Jeremy Corbyn reveal their rival visions for YorkshireElection 2017: May and Corbyn prepare for Yorkshire showdown in final pre-election TV grillingA statement issued by the leaders of France, Germany and Italy said the deal cannot be renegotiated, they remain committed to the "irreversible" accord and regard it as "a cornerstone in the co-operation between our countries, for effectively and timely tackling climate change".
In a phone call with the US president shortly after his White House announcement, Mrs May expressed her "disappointment" at the move and stressed the UK remains committed to the landmark 2015 agreement to curb greenhouse gas emissions.
But she faced fierce criticism for failing to add her name to the joint statement by President Emmanuel Macron, Chancellor Angela Merkel and Prime Minister Paolo Gentiloni.
Mr Corbyn labelled Mr Trump's move to withdraw from the Paris Agreement as "reckless and dangerous" and accused Mrs May of a failure of leadership in not signing up to the statement.
At an election campaign event in York, he said: "Given the chance to present a united front from our international partners she (Mrs May) has instead opted for silence and once again subservience to Donald Trump.
"It's a dereliction of both her duty to this country and our duty to our planet.
"This is not the type of leadership Britain needs either to negotiate Brexit or stand up to defend our planet in an era of climate change."
Liberal Democrat leader Tim Farron urged the Prime Minister to use whatever influence she has over the US president to urge him to reverse his decision.
"You have gone to Washington to hold Donald Trump's hand, now is the time to hold his feet to the fire," he said.
Scottish First Minister Nicola Sturgeon, who described Mr Trump's move as "profoundly regrettable", said it was an "appalling abdication of leadership by the PM" not to join her French, German and Italian counterparts in signing the joint declaration.
Green Party co-leader Caroline Lucas said Mrs May's "slow and timid" response to the US announcement was another sign of her weakness.
"Once again we're seeing a weak Prime Minister failing to stand up tall on the world stage in the face of near-criminal behaviour from one of our closest allies," she said.
In a statement, Downing Street said Mr Trump had called Mrs May to discuss his decision to withdraw from the agreement.
"The Prime Minister expressed her disappointment with the decision and stressed that the UK remained committed to the Paris Agreement, as she set out recently at the G7," the statement said.
"She said that the Paris Agreement provides the right global framework for protecting the prosperity and security of future generations, while keeping energy affordable and secure for our citizens and businesses."
Announcing his decision on Thursday, Mr Trump claimed the Paris deal allows countries such as China and India to carry on polluting while the US economy is harmed, complaining it is "less about the climate and more about other countries obtaining a financial advantage over the US".
He added: "We don't want other countries laughing at us any more, and they won't."
Mr Trump, who has previously labelled climate change a hoax by the Chinese to hurt US manufacturing, raised the possibility of negotiating to re-enter the Paris accord or an entirely new deal on terms that are "fair" to the US.
The accord commits countries to holding global temperature rises to "well below" 2C above pre-industrial levels, which will require global emissions to be cut to net zero by the second half of the century.
Scientists have warned failure to curb dangerous climate change will lead to sea level rises, more intense storms and flooding, more extreme droughts, water shortages and heatwaves as well as massive loss of wildlife and reduction in crop yields, potentially sparking conflict and mass migration.
Despite the decision by the US, the second biggest polluter after China, to pull out of the deal, many analysts suggest the shift to a low carbon economy is now unstoppable, with renewable prices tumbling and new clean technology being developed and deployed.
Mr Trump's decision prompted criticism from many US business leaders, including clean tech entrepreneur and Tesla chief executive Elon Musk and Robert Iger, chief executive of the Walt Disney Company, who said they had resigned from the president's advisory council over the issue.
A recent analysis of the commitments by countries under the Paris Agreement and the action they were taking showed a shift away from coal and boosting of renewable energy by China and India, which are set to overachieve on their targets.
The assessment by the Climate Action Tracker said China and India's action would slow global emissions growth, while moves to halt US policies on climate by Mr Trump were unlikely to have a major impact on global emissions by 2030.