Tehran has accused Washington of having "crossed all behavioral red lines of the worldwide community", by targeting judiciary chief Sadegh Amoli Larijani.
Iran also rejected any changes to its nuclear deal with world powers.
President Trump is re-certifying the Iranian nuclear deal Friday for another 120 days but warned it's the last time, as the USA seeks a tougher agreement against Tehran with European allies, and the administration slapped Iran with fresh sanctions.
"If other nations fail to act during this time, I will terminate our deal with Iran", he said.
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This would involve negotiations between the USA and its European allies rather than talks with Iran, the official said. These are separate to the nuclear deal.
He continued to waive sanctions against Iran on Friday, but said he was doing so "only in order to secure our European allies' agreement to fix the awful flaws of the Iran nuclear deal".
The waiver he will sign suspends sanctions for another 120 days but he warned that if a new agreement was not made, sanctions would not be waived again.
The nuclear deal between Iran and six world powers - the US, UK, Russia, France, China, and Germany calls upon Iran not to undertake any activity related to ballistic missiles created to be capable of delivering nuclear weapons, including launches using such ballistic missile technology.
Trump has argued that his predecessor, President Barack Obama, a Democrat, negotiated a bad deal for the United States in agreeing to the nuclear accord.
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"The deal is a long-lasting victory for Iran", Rohani said, referring to the accord Tehran signed in 2015 with the United States, Britain, France, Germany, Russia, and China. Under the current deal they are set to expire in 2025.
Iran must allow "immediate inspections at all sites requested by worldwide inspectors", he said, and "sunset" provisions imposing limits on Iran's nuclear programme must not expire.
"Rather than repeating exhausted rhetoric, U.S. must bring itself into full compliance - just like Iran", Mr Zarif tweeted immediately after Mr Trump's speech. "We are cutting off the regime's money flows to terrorists", the president said in the statement.
"This is the last time he'll issue waivers unless they reach an agreement", senior White House officials said.
President Donald Trump gestures as he talks to the media on South Lawn of the White House in Washington before his departure to Camp David last month.
Additionally, the president is hoping that the Iran deal will require better United Nations inspections of Iran's nuclear sites and that Iran avoid a "breakout period" of being able to produce enough uranium for a nuclear bomb.
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The EU said in a statement it had taken note of Trump's decision and would assess its implications. It prevents Iran from developing nuclear weapons while offering sanctions relief to allow the Islamic Republic to participate in global commerce and banking.