Trump said North Atlantic Treaty Organisation has been the "bulwark of global peace and security", and is no longer "obsolete".
Mr Trump has voiced caution on his relationship with Mr Putin, and said he would wait and see about future ties with Moscow.
"Frankly, Putin is backing a person that's truly an evil person", Trump told the Fox Business Network, referring to Assad.
In Paris, French President Francois Hollande said, "Russia bears a heavy responsibility" by systematically opposing multilateral action to protect its Syrian ally Assad.
Donald Trump has declared that USA relations with Russian Federation "may be at an all-time low" laying bare deep and risky divisions on Syria and other issues, while his Secretary of State Rex Tillerson offered a similarly grim assessment after meeting with Vladimir Putin in Moscow.
U.S. President Donald Trump has said the United States is "not getting along with Russian Federation at all" and that bilateral relations may be at an "all-time low".
The news conference came after Russian President Vladimir Putin met the top American diplomat for nearly two hours to see if they could rescue relations between the world's mightiest military powers.
"We have to see common threats" and not be guided by emotions, because "overthrowing the regime" may mean losing the battle against Islamic State, Lavrov said.
Russian Federation has vetoed a draft resolution at the UN Security Council that would have condemned last week's alleged chemical attack in Syria and demanded that Damascus cooperate with investigators.
The resolution, crafted by the United States, United Kingdom and France, was circulated to the 15 member countries of the Security Council on Tuesday.
First came last week's U.S. strike in response to Bashar al-Assad's deadly sarin-gas attack on innocent civilians. The Russians held to a tentative schedule that was worked out in advance, US officials said.
The day began with harsh words from Trump toward Putin, whom he had once praised effusively.
Tillerson's meeting with the Kremlin came amid heightened tensions between the US and Russian Federation, following America's strike on Syria, one of Russia's closest allies.
Back in Washington, Trump said: "We're not getting on with Russian Federation at all. They were there. So we'll find out", Trump said. "It's very bad for this world".
At his press conference following the meeting with NATO's secretary general, Trump said "it'd be a fantastic thing" if the two former Cold War foes were able to reconcile but warned that "it may not happen; it may be just the opposite".
The chemical attack killed at least 87 people, including many children, and images of the dead and of victims suffering from gas poisoning provoked global outrage. "I do not know who saw them".
Putin himself said earlier Wednesday that trust between the two countries had worsened since Donald Trump took office.
He added it was "highly likely" the Assad regime was behind the chemical attack, which prompted the United States to attack a regime airbase in retaliation.
"Right now, we're not getting along with Russia at all - we may be at an all-time low in terms of a relationship with Russia", Trump said Wednesday. As recently as March 30, his administration's stance, reinforced in individual statements by Secretary of State Rex Tillerson and Nikki Haley, ambassador to the United Nations, was not to intervene in Syria.
Russian Federation has criticized previous investigations carried out by the Organization for the Prohibition of Chemical Weapons and the United Nations which blamed the Syrian government for at least three chemical weapons attacks.
Russian Foreign Minister Sergey Lavrov blasted U.S. claims that it has "irrefutable evidence" of election interference.
Many watching the Trump-Russia storyline online agreed.
"I said it was obsolete; it's no longer obsolete", Trump said, standing beside Stoltenberg. Tillerson said in the news conference. "How do you know that?"