"This was a disgusting, one-sided deal that should have never, ever been made", President Trump said of the nuclear deal.
Mr Leilaz said he was not "worried about the remarks and tweets", and that "neither Iran, nor any other country is interested in escalating tensions in the region".
The speech, which came hours after Rouhani's threat but before Trump's tweet, has been interpreted as part of a concerted effort within the Trump administration to step up economic and political pressure on Iran.
"The president consults with his national security team on a daily basis", she said.
Press secretary Sarah Sanders says that "if anybody's inciting anything, look no further than to Iran".
Donald Trump wrote on Twitter: "To Iranian President Rouhani: NEVER, EVER THREATEN THE UNITED STATES AGAIN OR YOU WILL SUFFER CONSEQUENCES THE LIKES OF WHICH FEW THROUGHOUT HISTORY HAVE EVER SUFFERED BEFORE".
Were the United States to give their Israeli allies the green light to attack Iran directly, it is feared that Iranian proxies across the region, including Hezbollah and other deniable forces, could strike at American assets in the Gulf and the Middle East.
Rouhani, however, left open the possibility of peace between the countries, adding that the U.S.
In May, Trump announced that the United States is pulling out of the 2015 nuclear deal between Iran and six world powers and reimposing tough sanctions.
The withdrawal entailed not only the re-imposition of sanctions on Iran but also the so-called secondary sanctions on third countries.
1 dead, police arrest shooting suspect who was barricaded inside USA supermarket
Other videos showed an employee dragging a person out of the store and employees escaping on a rope ladder outside a window. Police suspect Atkins shot his grandmother seven times and wounded another woman before leading officers on the pursuit.
Although Iran and European signatories have been scrambling to save the deal, many Western companies and banks have stopped doing business in Iran amid fears they could be targeted by a reapplication of USA sanctions.
Israeli Prime Minister Benjamin Netanyahu, who has long pushed for a tougher stance against Iran by worldwide powers, praised Trump on Monday.
Willingness to negotiate at all with Iran is a substantial difference in Trump Administration position, however.
There is little interest among United States lawmakers for a conflict with Iran.
US President Donald Trump on May 8 chose to restore all the sanctions that had been lifted as part of a multi-national agreement, signed on to by former president Barack Obama's administration, in exchange for curbs on Iran's nuclear program.
Prominent Iranian analyst Seed Leilaz downplayed the war of words, describing it as "the storm before the calm".
Mr Trump's angry rhetoric has echoes of his Twitter barrages against North Korea's Kim Jong-un, whom he branded a "madman" who "will be tested like never before", before engaging in a testy exchange over whose nuclear button was bigger.
Iran's people want to be free from a corrupt government and also wish for closer ties with the United States, says a US senator.
The U.S. was a driving force behind the agreement under former President Barack Obama, but Trump has long taken issue with the accord, repeatedly calling it one of the "worst" deals he has ever seen.