President Trump, who has dismantled any diplomatic bridges that his predecessor Barack Obama built with Tehran, also warned the Iranian leadership: "We are no longer a country that will stand for your demented words of violence and death".
Pompeo on Sunday said Washington would resume sanctions against Tehran after Trump withdrew in May from a 2015 nuclear deal a handful of nations inked with Iran.
Mr Trump's furious tirade came after Mr Rouhani warned the U.S. via the country's state media that the "mother of all wars" would take place if the nation maintained its hostilities towards Iran.
"America should know peace with Iran is the mother of all peace, and war with Iran is the mother of all wars", the official Iranian government news agency IRNA quoted Rohani as saying at a July 22 gathering of Iranian diplomats.
Trump's eruption on Twitter came after a week of heavy controversy about Russian meddling in the US 2016 election, following the Helsinki summit with Russian President Vladimir Putin.
Mike Pompeo said the Iran regime has been a nightmare for its people. AFP
Tensions between Iran and the United States have worsened after Trump asked the long-handing nuclear deal created to ease sanctions in exchange for no nukes being developed by Tehran.
Pompeo branded Iran's leadership a "mafia" and promised U.S. backing to groups rebelling against the government, though did not name them.
"To our Iranian-American and Iranian friends: The Trump Administration dreams the same dreams for the people of #Iran as you do, and through our labors and God's providence they will one day come true", he tweeted.
Iran has rejected talks with the USA, and Rouhani has accused the US of stoking an "economic war". "USA!" The chaotic scene lasted for 30 seconds or so, with Pompeo eventually quipping, "If there were only so much freedom of expression in Iran", to cheers and applause.
Rouhani also suggested Iran could immediately ramp up its production of uranium in response to USA pressure.
Earlier this month, Israeli officials told reporters that the U.S. and Israel had already formed a "joint working group" created to plot against Iran, and that it's focus was to fuel internal unrest against the government. Norths Foreign Ministry accused the United States of making "gangster-like" demands for its unilateral disarmament.
Hundreds of 'White Helmet' rescue workers are evacuated from Syria
German Interior Minister Horst Seehofer said on Sunday giving the rescue workers shelter "is a humanitarian obligation. In 2017, a Netflix documentary about the group, " The White Helmets ", won the Oscar for best documentary short.