Since the nuclear accord was reached in 2015, the IAEA has carried out hundreds of inspections inside Iran.
He threatens "severe consequences" if Iran restarts its nuclear programme and calls instead for regime change.
"We will not allow American cities to be threatened with destruction", Trump said Tuesday from the White House Diplomatic Room, despite the fact that Iranian missiles can not reach the US.
Proponents of the deal have said those time limits were meant to encourage more discussion with Iran in the future that could eventually address other concerns. This was an agreement with Iran which resolved the long-standing difficulty which we've had for more than 10 years, where we saw Iran's nuclear program developing.
Trump's decision to withdraw from the accord and reimpose sanctions on the Islamic republic overturns years of diplomacy, could worsen instability in the Middle East and threatens business in Iran worth billions of dollars.
In this case, the Europeans are looking to defeat Mr. Trump's threatened sanctions. "At the drop of a hat, turning this deal around and retreating from this deal, is not just going to possibly impact the region but it might affect the entire world", Erdoğan told CNN International.
"It's very possible Iran may take the U.S.to this commission and say that the U.S.is violating this agreement", he said.
Mattis told lawmakers that President Trump did not make his decision in haste, suggesting that the administration is working on an alternative effort to prevent Iran from obtaining nuclear weapon capabilities.
For almost a decade, the nuclear question has crowded out serious deliberations over a broader policy toward Iran. I don't think they should do that. We may soon see that it is not Iran that is isolated from the world - it is the United States.
While that's not enough to satisfy Mr Trump it is enough to make the Iranians bridle at long-term European intentions. Washington has given European companies between 90 and 180 days to withdraw from Iran. "We can not prevent an Iranian nuclear bomb under the decaying and rotten structure of the current deal", he continued.
In this context, there is a massive distrust for Iran in the U.S. (and vice versa), and Washington has long feared what might happen if the Iranian regime developed a nuclear weapon.
He said the sanctions will sharply curtail sales of oil by Iran, which is now the world's fifth largest oil producer.
Rouhani for his part said Iran's economic interests would however need to be "guaranteed" in light of Trump's decision, the Iranian presidency's website said. If the US imposes sanctions on them, they've threatened "blocking" legislation that would make complying with those sanctions illegal.