In an interview to air Sunday on CBS' "Face the Nation", Tillerson said Russians were not targeted by the strikes.
The federation is facing the threat of fresh global sanctions over its support for the Syrian regime.
On North Korea's nuclear capability, Tillerson said Kim Jong Un "has made significant advancements in delivery systems, and that is what concerns us the most".
"We call on Russian Federation to do everything possible to bring about a political settlement in Syria and work with the rest of the worldwide community to ensure that the shocking events of the last week are never repeated", he said.
"By defeating ISIS and removing their caliphate from their control, we've now eliminated at least or minimized a particular threat not just to the United States, but to the whole stability in the region", he said.
"We deplore Russia's continued defence of the Assad regime even after the chemical weapons attack on innocent civilians".
And the Kremlin said Russian President Vladimir Putin and Iran's leader Hassan Rouhani condemned the cruise-missile attack Thursday on Syria's Shayrat airfield and urged the United Nations Security Council to take action against the USA for launching the attack that violated "the sovereignty of an independent country".
But he said the United States expected Russian Federation to take a tougher stance against Syria by rethinking its alliance with al-Assad because "every time one of these horrific attacks occurs, it draws Russian Federation closer into some level of responsibility".
In remarks from his Mar-a-Lago resort in West Palm Beach, Florida, President Donald Trump explained his decision to launch a military strike against a Syrian air base Thursday night, expressing disgust at the "very barbaric attack" by Syria.
Writing in The Sunday Times, Sir Michael said Russian Federation must be part of the solution to ending the "needless suffering" of Syrian civilians. Iran has also contributed Shiite militias to back Assad.
"I find it very disappointing but sadly I have to tell you not all that surprising", Tillerson added.
International Development Secretary Priti Patel said Britain will provide medical teams on the ground in Syria following the chemical attack.
Ms Patel said: "This was a brutal and indiscriminate attack".
"Those responsible must be held to account".
For his part, Rohani, whose country is a main backer of the Assad government, expressed his country's vehement condemnation to the US strike, stressing his Tehran's ongoing support to the Syrian government.