The United States will work with countries that import Iranian oil "to get imports as close to zero as possible" by November 4, Pompeo said.
It comes after Rouhani warned the United States against "playing with the lion's tail" and threatening Iran.
Mr Rouhani earlier said that war with Iran would be "the mother of all wars".
A headline on a local newspaper quoted Rouhani as saying: "Mr. Trump, do not play with the lion's tail".
Mr Rouhani has suggested Iran may block Gulf oil exports if its own trade is disrupted, a proposal backed by the country's supreme leader Ayatollah Ali Khamenei.
He says that unlike North Korea, "Iran never moved toward a nuclear bomb" and that therefore, "Iran is angry since Trump responded to Tehran's engagement diplomacy by pulling the USA out of the nuclear deal".
"Anyone who understands the rudiments of politics doesn't say "we will stop Iran's oil exports".we have been the guarantor of the regional waterway's security throughout history", Rouhani said, Reuters reported.
"America is threatening the Iranian people with its sanctions and our government is doing the same with its incompetence and mismanagement", said the self-employed 35-year-old.
Financial hardships have led many Iranians to take to the streets in protests, initially for economic reasons but often morphing into demonstrations against the government itself. "He is not in a position to act against Iran", Brigadier-General Qolam-Hossein Gheibparvar said on Monday. Trump's last message prior to his threat to Iran on Twitter was also in reference to President Obama.
Rouhani says conflict with Iran would be ‘mother of all wars’
Speaking in California, he said Iranian leaders had benefited from bribes and embezzlement.
Within hours, Iranian state-owned news agency IRNA dismissed Trump's tweet, describing it as a "passive reaction" to Rouhani's remarks. North's Foreign Ministry accused the United States of making "gangster-like" demands for its unilateral disarmament.
Many of the Iranian Americans in the audience either fled or are descendants of those who fled the country after the Islamic Revolution toppled the shah in 1979.
"The United States condemns the oppression levied on the Iranian people by those who rule unjustly", he said.
Several dozen protesters lined the route to the site, including one group that distributed fliers opposing both Iran's current government and any U.S. intervention in Iran.
Several members of the Trump administration and Trump's circle have been outspoken hawks on Iran.
"It framed the secretary's Reaganesque approach to confronting the Islamic Republic of Iran and his conviction that, with USA support, the demand of Iranians for freedom and democracy will leave the Islamic Republic on the ash heap of history as it left the Soviet Union", he said.
The 2015 agreement was in response to fears that Iran was developing a nuclear bomb.
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