Donald Trump's virulent speech contravened three articles of the 2015 deal, Mohammad Javad Zarif said in televised remarks broadcast late on Saturday.
The clauses he referred to mandate the signatories to implement the accord "in good faith" and the US administration and Congress to "refrain from re-introducing or re-imposing" sanctions related to Iran's nuclear program.
"I have already written nine letters (to EU foreign policy chief Federica Mogherini) listing the cases where the United States has failed to act on or delayed its commitments under the JCPOA," Mr Zarif added, calling the nuclear deal by its technical name.
Mr Zarif said he was going to write a new letter regarding Mr Trump's speech on Friday.
He called on the US to prolong sanctions relief, otherwise "the Islamic Republic of Iran will definitely take a similar and reciprocal measure".
In his speech, Mr Trump refused to "certify" Tehran's compliance with its obligations under the nuclear deal.
He warned he could rip up the agreement "at any time", saying it had failed to address Iran's influence in its region and its illegal missile programme.
Mr Trump said he supported efforts in Congress to work on new measures to address these threats.
The remarks triggered swift condemnation from Iranian President Hassan Rouhani of what he called "baseless accusations and swear words".
Responding to US threats of new sanctions against Tehran's missile programme, Mr Zarif repeated Iran would not tolerate interference in its defence policy.
"Our achievements in the field of ballistics are in no way negotiable," he said.
"We live in a region into which hundreds of billions of dollars of lethal American weapons have poured, turning it into a gunpowder storehouse... so we have the right to have defensive means," he said.
The nuclear agreement was signed in July 2015 between Iran and six world powers - Britain, China, France, Germany, Russia and the United States.
Ms Mogherini helped to negotiate the nuclear deal.
It lifted some Western and UN sanctions and in return Tehran curtailed a large part of its atomic programme to assure the world that a nuclear bomb - which Iran denies seeking - would remain well out of its reach.
