Secretary of State Rex Tillerson met his Russian counterpart Foreign Minister Sergei Lavrov in Moscow in what he said was an attempt to narrow "areas of sharp difference".
Earlier on Wednesday Tillerson had a similar tone after an nearly two-hour meeting with Putin, saying the two countries had reached a "low point" in relations.
Mr Johnson said: "This afternoon in NY, the global community sought to make clear that any use of chemical weapons by anyone anywhere is unacceptable and that those responsible will face consequences".
"The world's two foremost nuclear powers can not have this kind of relationship", he said.
He also rejected American claims of certain evidence that Assad ordered the chemical attack.
Lavrov on Friday expressed skepticism about a preliminary investigation conducted by the U.N.'s chemical weapons watchdog.
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One of his deputies went further, telling Russia's state-owned news agency: "In general, primitiveness and loutishness are very characteristic of the current rhetoric coming out of Washington". Assad referred to the aftermath of the attack seen on social media and in the news as a "play". -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.
Still, Mr Lavrov sought to put a positive spin on the talks with Mr Tillerson, saying they helped improve mutual understanding.
The Kremlin had previously declined to confirm Putin would meet Tillerson, reflecting the renewed tensions.
Tillerson, who arrived on Tuesday on his first visit to Russia since he was named to lead the U.S. State Department, is meeting for talks with Russian Foreign Minister Sergey Lavrov. "Russia, as their closest ally in the conflict, perhaps has the best means of helping Assad recognize this reality".
Lavrov repeated the Kremlin's condemnation of the U.S. missile strike in Syria and said Washington was seeking "excuses for regime change".
"There was no order to make any attack, we don't have any chemical weapons, we gave up our arsenal a few years ago", Assad said, adding: "Even if we have them, we wouldn't use them, and we have never used our chemical arsenal in our history".