President Donald Trump said the strikes were a response to the chemical weapons attack carried out against Syrian civilians on Tuesday, for which the United States blames the Syrian government.
The Moscow news conference came after Putin met Tillerson for the first time since Trump took office.
Trump said from the White House's East Room that he and UN Secretary General Jens Stoltenberg had a "productive" discussion this afternoon about the alliance and the aid in can provide in the global fight against terrorism.
And the rhetoric from Moscow had been upped also.
Russia's veto was the eighth by the country at the UN Security Council, in a bid to shield Syrian President Bashar al-Assad.
At the White House press conference, Trump, who has repeatedly questioned NATO's objective, also said the Western military alliance is "no longer obsolete".
During his visit to Moscow on Wednesday, Mr Tillerson said US-Russia ties had reached a "low point" - despite a lengthy meeting with Mr Putin and Mr Lavrov.
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"The world s two foremost nuclear powers can not have this kind of relationship", Tillerson said. Tillerson is considering the request, he said, "and we expect that Washington will have a constructive reaction".
Further punctuating the Syria dispute, Russian Federation on Wednesday vetoed a Western-backed resolution at the United Nations Security Council that condemned the chemical weapons attack.
US President Donald Trump told allies on Wednesday it was time to end Syria's "brutal" civil war.
The presidents of the United States and Russian Federation on Wednesday both presented souring views of the relationship between their two countries, exchanging sharp words as Moscow extended an icy welcome to the United States' top diplomat in a face-off over Syria.
The vote came the day after Trump spoke by phone to President Xi Jinping of China, whom he hosted last week at a summit at his Mar-a-Lago retreat in Palm Beach, Florida. "It is not too late for Russian Federation to fulfill its responsibilities as a permanent member of this Security Council", he said.
Trump expressed hope that he can improve relations with Putin, adding, "We're going to see what happens".
"As to the question of the interference with the election, that is fairly well established in the United States and I think that has been spoken to on the Hill as well with the Congress. It's very bad for this world".
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China, which has also vetoed several resolutions in the past along with Russian Federation, abstained on this one, a move welcomed by President Donald Trump at a White House news conference.
Asked about Russia's prior knowledge about the Syrian government's alleged use of chemical weapons, Trump said it's "possible", but "unlikely" that Russian Federation had known in advance of Syria's plan to launch a chemical weapons attack on its own citizens. "It'd be a fantastic thing if we got along with Putin and if we got along with Russian Federation, and that could happen, and it may not happen; it may be just the opposite". Still, he held out hope that the United States and Russian Federation could come to terms, suggesting that Tillerson's talks with Putin had gone better than expected. The U.S. also accused Russian Federation of mounting a disinformation campaign aimed at exonerating Assad from blame for the attack.
But Russia made clear it had no intention to acquiesce. Of course, even limited engagement by the Trump administration could significantly strengthen the Free Syrian Army (FSA) and create problems for Russia, Iran and the regime.
"It's really pitiful to hear such irresponsible and provocative statements from the most important leader in the Western world", Ja'afari said. "Our impression is that the West, mainly the United States, is hand-in-glove with the terrorists".
Still, the turnabout drew praise from some lawmakers who had been concerned with Trump's previous stance.
Trump said Tillerson had completed a successful meeting with Putin, where "things went pretty well". "Nato is the most successful military alliance in human history".
US envoy Rex Tillerson meets President Putin in Moscow
Secretary of State Rex Tillerson shakes hands with Russian Foreign Minister Sergei Lavrov at the start of their meeting in Moscow. Until the chemical attack, the Trump administration had sought to step back from the US position that Assad should leave power.