The State Department's Stuart Jones appeared took a 20-second-long silent pause when a reporter asked him about U.S.' backing of democracy and Saudi Arabia
Muslim-majority countries have different ways of determining the start of Islam's holiest month.
Speaking in Jerusalem the U.S. president also said Iran needed to cease funding of terror organizations.
Tehran and Riyadh are regional arch-rivals which accuse each other of sponsoring fundamentalist militias aligned to their competing sects of Islam.
Rouhani had a resounding win but angered hardliners with his comments during the campaign, and they are likely to create problems for him.
Syrian and Iranian concerns regarding the U.S. and Jordan's nearby military drills were heightened when U.S. troops were seen with rebels in Syria.
U.S. officials say Iran recently tested its supercaptivating Hoot torpedo, capable of traveling 250 miles per hour by evaporating the water around it.
The industrial disaster in north-eastern Iran is the most deadly in the Islamic Republic for decades.
Iran's defense minister was responding to Saudi comments about the "battle" for influence in the Middle East.
The Syrian government says it destroyed its stockpiles in 2014.
New footage emerged of Iran's Qaher F-33 stealth fighter jet, which Western critics said was further away from its first flight than Tehran claimed.
The organization is seen as a counterweight to Western alliances, cooperating on political and economic affairs.
The evacuation of civilians in four besieged Syrian towns is also part of the deal.
Experts say a Raisi victory in the presidential elections could increase already simmering tensions between the Islamic Republic and the West.
The minister took a shot at the Islamic Republic while commenting on the murder of a top Hamas member.