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6 year oldSenior Russian figures had threatened to meet any US strikes with a response.
Mr Trump had promised a "forceful" reply to the suspected attack.
President Bashar al-Assad's government, which receives military backing from Russia, denies being behind any chemical attack.
In his tweet, Mr Trump called Mr Assad a "gas killing animal".
On Saturday, Syrian opposition activists, rescue workers and medics said the rebel-held town of Douma in the Eastern Ghouta region had been attacked by government forces using bombs filled with toxic chemicals.
The Syrian-American Medical Society said more than 500 people had been found with symptoms "indicative of exposure to a chemical agent", and on Wednesday the World Health Organization demanded access to the area to verify reports from its partners that 70 people had died.
After the attack, Mr Trump and his Defence Secretary, James Mattis, cancelled travel plans.
The US and Western allies said they have agreed to work together.
But several senior Russian figures warned of a Russian response to a US attack, with Alexander Zasypkin, Moscow's ambassador to Lebanon, repeating on Wednesday a warning by the head of the military that missiles would be shot down and their launch sites targeted.
Mr Trump's tweet referred to this warning.
Russia vows to shoot down any and all missiles fired at Syria. Get ready Russia, because they will be coming, nice and new and “smart!” You shouldn’t be partners with a Gas Killing Animal who kills his people and enjoys it!
— Donald J. Trump (@realDonaldTrump) April 11, 2018
The US and allies - France and Britain - are believed to be preparing for a military strike.
US President Donald Trump cancelled his first official trip to Latin America so he could focus on Syria.
That decision suggests the US response may involve a larger military operation than a limited strike, says the BBC's Barbara Plett Usher in Washington.
French President Emmanuel Macron said any strikes would target Syrian government chemical facilities.
But The Times reports that the UK's Prime Minister Theresa May has urged Mr Trump to provide more evidence of the suspected chemical attack.
A US Navy guided-missile destroyer, the USS Donald Cook, is in the Mediterranean, and the European air traffic control agency, Eurocontrol, has warned airlines to take "due consideration" while in the eastern Mediterranean over the next few days, because of the possible launch of missiles into Syria.
Moscow's UN ambassador, Vasily Nebenzia, warned Washington that it would "bear responsibility" for any "illegal military adventure" it carried out.
On Tuesday the UN Security Council failed to pass any measures to set up an inquiry into the alleged attack.
As permanent members of the council, Russia and the US both have the power of veto. They both blocked each other's proposals to set up independent investigations.
The US-drafted resolution would have allowed investigators to apportion blame for the suspected attack, while Russia's version would have left that to the Security Council.
A team from the Organisation for the Prohibition of Chemical Weapons (OPCW) is due to deploy to Syria "shortly" to determine whether banned weapons were used in Douma.
But the OPCW will not seek to establish who was responsible for the attack.
The UN session was the latest in a series of showdowns between Russia and the US and saw harsh words exchanged between the countries.
Ambassador Nebenzia accused the US of "planting" a resolution" as a justification for military action.
"We could find ourselves on the threshold of some very sad and serious events," he said.
US envoy Nikki Haley responded by calling the vote a "travesty".
"Russia has trashed the credibility of the council," she said. "Whenever we propose anything meaningful on Syria, Russia vetoes it."
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