He said this while talking to his Syrian counterpart Walid al-Moualem in Moscow.
President Donald Trump declared that US-Russia ties "may be at an all-time low", and Mr Tillerson struck a similar tone after a day of talks in Moscow.
He added there was no talk on Wednesday about a possible Trump-Putin meeting.
The U.S. missile strike on Syria that triggered the diplomatic crisis hit a Syrian airbase where there had been Russian military personnel, so did Moscow know about the chemical weapons? "I think he had a very successful meeting", Trump said.
President Donald Trump said it would be "wonderful" if NATO and the United States could get along with Russian Federation during a news conference with NATO Secretary General Jens Stoltenberg.
The result: not only no warm up in relations between the two superpowers, but, as Tillerson put it, "a low level of trust between our countries", a relation that "the world's two foremost nuclear powers can not have". Russia wants the United States to eliminate sanctions on Moscow related to its 2014 annexation of Ukraine's Crimea region and support for pro-Russian separatists in eastern Ukraine. First and foremost, there was an important conversation with President Vladimir Putin.
Russia Wednesday vetoed a Western draft Security Council resolution on an alleged chemical attack in Syria's northwestern province of Idlib.
This comes after consensus from USA and world leaders that his government forces were behind the attack.
"Where is the proof that Syrian troops used chemical weapons?" "Such attempts won't succeed", Lavrov said.
"We have not seen a single fact, or even a hint of facts", he said. So that we can then address the more serious problems. "That's a butcher", Trump said.
While the most immediate US-Russian dispute concerns culpability for the chemical weapons, broader disagreements over everything from Ukraine to Russia's support for once-fringe candidates in European elections are among other sore points.
Tillerson was greeted frostily in the Russian capital as Lavrov began their meeting Wednesday by demanding to know America's "real intentions". "Foreign Minister Lavrov and I agreed we would consider further proposals made about the way forward in Syria, including consulting with our allies and coalition members and we will continue discussions about how to find a solution to the Syrian conflict". "We urge the USA and their allies to respect Syria's sovereignty".
Russia's hopes for a thaw in Russia-US ties following Mr Trump's election have been shattered by the congressional investigation into alleged ties between Trump campaign associates and Russia.