Rouhani was elected in June promising a more diplomatic approach to the West in negotiations over its nuclear program, which the U.S. and its allies say is aimed at weapons development. Iran says the program is for peaceful purposes.
The new president has stated however that Iran will not give up its right to disputed nuclear activities, nor has he suggested any major rethinking of Iran's alliance with Damascus. Key nuclear and security matters are under the control of the country's Supreme Leader Ayatollah Ali Khamenei and institutions that answer directly to him, including the Revolutionary Guard.
Syria's opposition says the Guard is directly involved in the war on Assad's side. Iran denies this.
"The Guards do not seek military domination over the region," Rouhani said. "It is a mistake by the West, which thinks Iran is after military domination over the region."
He called on the U.S. military, which has bases in the Persian Gulf, to pull out of the region. "You, the unsolicited guest in our region, leave the region and then you will see that it becomes heaven."
On Monday Iran's Foreign Minister Mohammad Javad Zarif said his country is ready to "resolve the nuclear issue, if the West is serious." He said Tehran was willing to "build trust" with Washington.
"We are suffering from lack of mutual trust," Zarif told Lebanese news TV channel al-Mayadeen.
Rouhani and Zarif plan to attend the U.N. General Assembly in New York later in September. Rouhani has suggested it may be an opportunity to move forward in nuclear talks.
In the same meeting, Guard commander Gen. Mohammad Ali Jafari said, "So far, the enemies' plot for military intervention in Syria has failed. We are hopeful this will continue until the end."