On Tehran streets, residents took the exchange in stride.
Iran officials have threated to close the Hormuz Strait in the past in relation to any potential attacks.
The high-stakes verbal sparring is reminiscent of the exchanges Trump had previous year with North Korean leader Kim Jong Un before the two leaders met in a historic summit last month.
Meanwhile, Trump's tweet was reverberating across the Mideast.
Regularly suspected of favouring regime change in Iran, Pompeo refused to distinguish between moderates and radicals at the heart of the Islamic republic.
The sanctions are set to come into play after Trump's decision to withdraw the United States from a 2015 global deal over Iran's nuclear program.
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Trump has been a vocal critic of Iran since before he took office, vowing to scrap the Obama-negotiated nuclear deal - which he did in May, despite allies including the United Kingdom highlighting its success so far. Rouhani also said that a peace with Tehran is like "the mother of all peace".
Rouhani had also affirmed that the country will continue to have economic ties with other nations despite the US-imposed sanctions.
Pompeo's speech drew criticism from domestic critics of the administration, including veteran diplomats behind the landmark Iran nuclear deal.
Trump's tweet suggested he has little patience with the trading of hostile messages with Iran, using exceptionally strong language and writing an all-capitalized tweet. "BE CAUTIOUS!", he wrote.
There have been a series of anti-government protests in Iran in recent months. The audience included Iranian-Americans, Arkansas Senator Tom Cotton - a leading Iran critic in Congress - and former California Governor Pete Wilson.
In the case of North Korea, the public war of words cooled quickly and gradually led to the high profile summit and denuclearisation talks.
Iranian lawmaker Heshmatollah Falahatpisheh told The Associated Press that he doubted it would come to a military confrontation between Iran and the United States, despite the escalating rhetoric. The United States supports you.
The European Union is likely to be hit by U.S. sanctions - although South Korea foreign minister Kang Kyung-Wha, speaking at a breakfast seminar on Thursday prior to flying to NY for talks with Mr Pompeo, said she was hopeful of securing a waiver which would allow her country to continue buying oil from Iran.