Tillerson's more restrained language may reflect his plans to visit Moscow for meetings this week, however, where he hopes to bridge the latest tensions in ties.
In a phone call with Syrian counterpart Bashar al-Assad, Rouhani told him, "Allegations that Syria launched the chemical attack were just a pretext to disrupt the Syrian peace process", according to Press TV.
Meanwhile, Italy wants the group to issue a communique that will reinforce the United Nations -backed peace process in the war-torn country, Reuters reported.
Haley also noted that the United States' first priority is still to defeat ISIS, but that there can be "multiple priorities".
The top Russian and Iranian generals have condemned the USA missile strike against a Syrian military base and vowed to continue their fight against "terrorists", which they generally call opponents of Syrian President Bashar al-Assad.
Sen. John McCain on Sunday lobbed part of the blame for Syria having used chemical weapons against its own citizens at President Trump and his administration.
"We have information from our partners and from contacts on that ground that makes it very plausible that the Assad regime was behind this disgusting poison gas attack", he said.
The Syrian regime and its Russian and Iranian backers have vowed to keep up the fight against "terrorists" despite threats of further U.S. strikes against the regime following the chemical attack.
Grand National hero Derek Fox: I still can't believe it
SCOTTISH racing fans were enjoying a betting bonanza last night after the country's first Grand National win in 38 years . He added: "The racecourses here in Scotland, they are the best administered, the best racecourses I've ever come across".
Russian Federation had agreed to ensure Syria's chemical weapons stockpiles were destroyed - and its failure to do this enabled the attack, he said.
"The failure related to the recent strike and the recent awful chemical weapons attack in large measure is a failure on Russia's part to achieve its commitment to the worldwide community", he added.
A total of 87 people, including many children, died in the suspected sarin attack on the rebel-held town of Khan Sheikhoun in Idlib province on Tuesday.
Responding to Russia's claim that the United States missile strike on a Syrian airbase wasn't very effective, President Donald Trump has explained as to why the runways were not attacked and remain undamaged.
"We are prepared to do more", he said.
WATCH: Syria Strike: Finally some action or unsafe reaction?
But the regime has denied any responsibility, saying its air strikes hit a rebel depot containing chemical materials.
More than 320,000 people have been killed in Syria since the conflict began in March 2011 with a crackdown by Assad against antigovernment protesters.