Trump earlier denounced the "rotting and decaying" agreement, which committed the US to easing a series of sanctions in the Islamic Republic.
The treaty – officially known as the Joint Comprehensive Plan of Action (JCPOA) – took years to develop and saw the Middle Eastern nation agree to reduce its stockpile of medium-enriched uranium by 98 per cent.
But today Trump claimed Iran had continued to develop its nuke programme, saying: "We have definitive proof that this Iranian promise was a lie."
Iranian President Hassan Rouhani hit back in a defiant press conference in Tehran.
He told reporters: "I have ordered Atomic Energy Organisation of Iran to be ready to start the enrichment of uranium at industrial levels.
"We will wait a few weeks and speak with our allies and friends and those committed to JCPOA.
"All depends on our national interests."
The US withdrawal from the treaty, which was brokered by seven parties and signed in 2015, will place everyone in uncharted territory.
President Rouhani was defiant in his speech, saying his country would remain in a nuclear deal without the US.
He said: "Iran is a country clear of its commitment and the US is a country that has never adhered to its commitments.
"Our history over the past 40 years has told us that the Americans speaking of their approach they've always adopted a hostile approach.
"Tonight they announced they have pulled out of the JCPOA and in fact that was an emphasis on what they have been doing.
He added: "If JCPOA turns into a piece of paper without guaranteeing our interests there will be a clear path ahead for us.
"I assure the people of Iran we will overcome this issue."
Enriched uranium is a critical component for making nuclear weapons and in nuclear power stations and by curbing the amount Iran produce is a way to curb the number of weapons produced.
As part of the agreement Iran also agreed to only enrich their uranium up to 3.67 per cent over the next 15 years and they agreed to reduce their gas centrifuges for 13 years.
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