He said the US would "apply unprecedented financial pressure" to bring Tehran back to the table.
Pompeo said if Iran were to abide by the stricter terms, including ending its ballistic missile program and its interventions in regional conflicts from Yemen to Syria, the United States would lift its new sanctions.
Washington appeared to be planning to fold all of its issues of contention with Tehran into "a giant negotiation, a new jumbo Iran negotiation, a new treaty", said the Foreign Secretary.
Total signed a contract in 2017 to develop phase 11 of Iran's South Pars field with an initial investment of $1billion.
Israeli PM Praises Pompeo's Iran Speech as 'Policy That Can Guarantee Security'
Zarif says that the EU's support of the deal is not in step with the threat made by some major European companies to withdraw cooperation with Iran.
In an effort to save the deal, the EU will, The Hill noted, invoke the so-called blocking statute, meant to protect European companies doing business with Iran from American sanctions.
Pompeo also demanded that Iran cease from a range of activities throughout the Middle East that have long drawn the ire of the USA and its allies.
Iran's foreign minister has said that the nation will not renegotiate the 2015 Iran nuclear deal.
Pompeo also warned European businesses who have signed business deals in recent years to work with Iran that if they are in violation of US sanctions they will be held "to account". "Every country's going to have to understand that we can not continue to create wealth for [Quds Force Commander] Qasem Soleimani", Pompeo said.More news: The Royal Wedding: Prince Harry, Meghan Markle, And The Guests
"For two generations the Iranian regime as exacted a heavy toll on its own people and the world", Pompeo said.
Al Jazeera's Zein Basravi, reporting from Tehran, said that despite Pompeo's strong rhetoric, the Iranian leadership is "likely to feel like this is nothing they haven't heard from the United States before".
He added: "We will track down Iranian operatives and their proxies around the world and we will crush them".
When the full force of the sanctions are in place, Iran "will be battling to keep its economy alive", forcing Iran to choose between that or squandering resources overseas.More news: Priyanka Chopra sports chic dress suit at royal wedding
"It will not have the money to do both", he said.
At the same time, Mr Pompeo offered Iran a series of dramatic potential USA concessions if it agrees to make "major changes".
Several global energy companies say they are also reconsidering their current projects in Iran.
Taken together, the demands would require a complete transformation by Iran's government, and they hardened the perception that the administration is really seeking regime change. "We long to see them prosper and flourish as in decades past, and as never before". "And we'd be willing to support the reintegration of Iran's economy in the worldwide economic system".
There was no immediate comment from Iran to Pompeo's speech, but after Washington's exit, Tehran had said that "the interests of the people of Iran must be assured" for it to stay in the deal.More news: Next CW superhero crossover to introduce Batwoman
Pompeo did offer Iran a series of dramatic potential US concessions if it agrees to make "major changes".