Update: Britain, France and Germany "stand committed" to the Iran nuclear deal and are "concerned by the possible implications" of Donald Trump's refusal to back it, Theresa May, Angela Merkel and Emmanuel Macron said in a joint statement.
The US president accused Tehran of violating the spirit of the landmark 2015 agreement and is referring it to Congress.
Mrs May, the German Chancellor and French president said preserving the pact was "in our shared national security interest" and called for Washington to "consider the implications" of taking action that undermines it.
Mr Trump stopped short of ripping up the deal but said without measures to toughen it up "the agreement will be terminated".
The statement from the UK, France and Germany said the International Atomic Energy Agency (IAEA) had "repeatedly confirmed" Iran's compliance to the terms it signed up to.
It said: " We, the leaders of France, Germany and the United Kingdom take note of President Trump's decision not to recertify Iran's compliance with the Joint Comprehensive Plan of Action (JCPoA) to Congress and are concerned by the possible implications.
"We stand committed to the JCPoA and its full implementation by all sides. Preserving the JCPoA is in our shared national security interest. The nuclear deal was the culmination of 13 years of diplomacy and was a major step towards ensuring that Iran's nuclear programme is not diverted for military purposes."
It added: "We encourage the US administration and Congress to consider the implications to the security of the US and its allies before taking any steps that might undermine the JCPoA, such as re-imposing sanctions on Iran lifted under the agreement.
" At the same time as we work to preserve the JCPoA, we share concerns about Iran's ballistic missile programme and regional activities that also affect our European security interests.
"We stand ready to take further appropriate measures to address these issues in close cooperation with the US and all relevant partners. We look to Iran to engage in constructive dialogue to stop destabilising actions and work towards negotiated solutions."
Earlier: President Donald Trump has angrily accused Iran of violating the spirit of the landmark 2015 nuclear deal but stopped short of ripping up the agreement.
In a speech at the White House, the US president announced a new strategy, saying the administration would seek to counter the regime's destabilising activities and would impose additional sanctions to block its financing of terrorism.
But Mr Trump said he was not yet ready to implement a campaign pledge to pull the US out of the deal or re-impose nuclear sanctions.
Instead, he moved the issue to Congress and the other parties to the seven-nation accord, telling legislators to toughen the law that governs US participation and to fix a series of deficiencies in the agreement. Those include the expiration of several key restrictions under "sunset provisions" that begin to kick in in 2025, he said.
Mr Trump warned that without the fixes, he was minded to pull the US out of the deal and snap previously lifted sanctions back into place.
Without improvements, he said, "the agreement will be terminated".
"It is under continuous review and our participation can be cancelled by me as president at any time," he said.
House Speaker Paul Ryan backed Mr Trump's decision to re-examine the seven-nation accord which he claimed was "fatally flawed".
The Wisconsin Republican said weaknesses in the nuclear agreement would allow Iran "to pursue nuclear weapons under the guise of international legitimacy" once specific restrictions on Iran's nuclear program expire after predetermined periods of time.
He warned simply enforcing a bad agreement was not sufficient.
Mr Trump's announcement was essentially a compromise that allows him to condemn an accord that he has repeatedly denounced as the worst deal in American history. But he stopped well short of torpedoing the pact, which was negotiated over 18 months by the Obama administration, European allies and others.
Congress will now have 60 days to decide whether to put the accord's previous sanctions back into place, modify them or do nothing. Any decision to re-impose sanctions would automatically kill America's participation in the deal.
After his announcement Mr Trump said that if Congress doesn't come up with satisfactory changes to the deal in a "very short" period of time, then he's prepared to "terminate" it.
He said he was "very unhappy with Iran" and warned that the country "has to behave much differently".
Mr Trump defended his idea for Congress to come up with a fix.
"I like the two-step process much better," he said.
Still, he said, he may choose to pull out at a later date: "We'll see what happens over the next short period of time."
Ahead of Mr Trump's speech, Secretary of State Rex Tillerson and other US officials offered details of the new stance.
Mr Tillerson said Mr Trump wanted lawmakers to come up with legislation that would automatically re-impose sanctions that were lifted under the deal should Iran cross any one of numerous nuclear and non-nuclear "trigger points."
Those "trigger points" would include violations of the deal involving illicit atomic work or ballistic missile testing, support for Syrian President Bashar Assad, Lebanon's Hezbollah movement and other groups that destabilise the region, human rights abuses and cyber warfare, Mr Tillerson said.
Both defenders of the Iran nuclear deal and critics are likely to be displeased by Mr Trump's decision. Those who support the deal believe the move will badly damage US credibility in future international negotiations, while opponents think he does not go far enough in unravelling the accord.
Ali Larijani, Iran's parliament speaker, said that any US move against a nuclear deal with Iran would be an "insult" to the United Nations which had given the deal its blessing.
He added that any revision of the deal would allow Iran to take its own actions, and warned that the US move could destabilise the international situation.
"We will continue to adhere to our obligations ... for as long as other parties observe the agreement," he said on a visit to Russia.
American allies, who have pressed the White House to remain in the nuclear accord, were closely watching the president's address.
The European parties to the accord - Germany, France and Britain - along with the other parties, Iran, Russia and China, have ruled out reopening the deal. But some, notably France, have signalled a willingness to tackle unresolved issues in supplementary negotiations.
Among those issues are the expiration of several restrictions on advanced nuclear activity under so-called "sunset clauses" that will allow Iran to begin ramping up its enrichment capabilities after 10 years, the end of an arms embargo and the eventual easing of demands for a halt to its missile program.
In the speech, Mr Trump hoped to "recruit" the Europeans into joining his broad strategy, particularly by punishing the Revolutionary Guard, which he and his national security team believe is fomenting instability, violence and extremism throughout the Middle East and beyond, according to one official.