Former House speaker Gingrich and ex-New York mayor Giuliani also told a rally of thousands of Iranian opposition supporters in Paris that Trump needed to turn up the heat on European countries still seeking to do business with Tehran despite reimposed US sanctions.
“The only way to safety in the region is to replace the dictatorship with a democracy and that has to be our goal,” Gingrich told the Free Iran rally, organised by exiled opponents including the former rebel People's Mojahedin which is banned in Iran.
He stressed he did not speak for the Trump administration, but added: “It seems to me there would be a rather happy celebration should regime change occur.”
Gingrich said he did not support arming the Iranian opposition, saying Trump should instead heap on more sanctions after pulling out of the 2015 nuclear deal between Tehran and world powers.
And he blasted countries attempting to find ways to allow their companies to keep operating in Iran under the threat of penalties for US sanction-busting.
“We need to have a campaign to shame the European governments who are unwilling to support freedom and democracy,” Gingrich said. “We need to insist that they join the sanctions once again.”
Giuliani called for a boycott of companies “that continually do business with this regime”.
“Freedom is right around the corner,” he added of the recent protests in Iran.
Gingrich, Giuliani and other US politicians have been heavily paid to speak at the annual Paris rally in recent years.
Their comments came after US Secretary Mike Pompeo this week gave his backing to the strikes and protests over economic woes, not least the collapse of the currency following the US withdrawal from the nuclear deal. On Monday traders at Tehran's Grand Bazaar staged a rare strike following earlier reports of street protests in provincial cities.
Iranians have been hit by rising prices, and record levels of unemployment have left a third of under 30s out of work.
The latest protests follow dozens over the new year which left at least 25 people dead.
The People's Mojahedin were founded in the 1960s against Iran's royalist government and went on to fight the Islamic regime after the 1979 revolution.
In 2009 the European Union struck the Mojahedin from its list of terrorist organisations, where it had been since 2002. The United States did the same in 2012.