Tillerson made the remarks during a joint news conference with his Saudi counterpart on Saturday following Rouhani’s landslide election victory and winning a second term in office through presidential polls held on Friday.
He repeated an old American accusation that Iran supports terrorism, saying, "What I would hope, is that Rouhani... use that (new) term to begin a process of dismantling Iran's network of terrorism, dismantling its financing of the terrorist network, dismantling the manning and the logistics and everything that they provide to these destabilizing forces that exist in this region."
"We also hope that he puts an end to their ballistic missile testing," Tillerson said on the first day of a visit by President Donald Trump to Saudi Arabia where he signed a whopping $110 billion weapons deal with the Saudi kingdom.
Tehran has always said its missile tests pose no threat to any country and that they are conducted to boost Iran’s defense capabilities, enhance national security and promote regional peace and stability.
The United States and its allies, particularly Saudi Arabia, however, have always drawn massive criticism for supporting and financing terrorist groups, which have carried out numerous terrorist attacks in the Middle East, throughout Europe and elsewhere.
Since 2011, the Saudi regime has also been sponsoring Takfiri terrorists fighting against the Syrian government, which has left hundreds of thousands dead and millions more displaced.
Trump, who arrived in Riyadh on Friday, signed an arms deal with the Saudi kingdom that has been pounding Yemen since March 2015, an assault that has so far left 12,000 people dead, most of them civilians.
The pact includes a $6 billion contract to assemble 150 Lockheed Martin Blackhawk helicopters in Saudi Arabia. It also includes a $1 billion THAAD missile system and contract for four multi-mission warships worth $11.5 billion.