Nikki Haley, the US ambassador to the United Nations, said the White House warning was meant for a wider audience.
Syria appears to have heeded a warning against staging any new chemical weapons attack as no such action has been launched, US Defence Secretary Jim Mattis said on Wednesday. On Monday, the White House warned that Syrian forces would "pay a heavy price" if they carried out another chemical strike.
The security official said if the U.S.is right it must allow the Organization for the Prohibition of Chemical Weapons (OPCW) investigate the issue.
The U.S. military says it will assess allegations that a coalition airstrike may have killed over 40 prisoners held in an Islamic State-run jail in eastern Syria. President Donald Trump previously ordered an April 7 cruise missile strike on one of Assad's airfields after a sarin gas attack on civilians.
Trump has said he won't stand for Syria's use of chemical weapons, which are banned under worldwide law and are particularly worrisome in the Arab country because they could fall into extremists' hands.
The warning came after U.S. intelligence noticed suspect activity at the airbase used to launch a suspected chemical strike two months ago.More news: Former Baltimore Ravens linebacker Zach Orr considering comeback
Syria's two main allies, Russian Federation and Iran, joined in criticizing the U.S. Iran's foreign minister called the U.S. threat a "dangerous escalation".
The situation at Shayrat airfield has not changed.
Nikki Haley, the U.S. Ambassador to the United Nations, followed up Spicer's statement with a Twitter warning: "Any further attacks done to the people of Syria will be blamed on Assad, but also on Russia & Iran who support him killing his own people".
Also, a senior Russian lawmaker on Tuesday dismissed the USA claim as an "unprecedented provocation".
The U.S. strike was the first direct American assault on the Syrian government and Trump's most dramatic military order since becoming president months before.
But then U.S. president Barack Obama, who had also declared that a chemical attack would cross a "red line", eventually decided against military action. Last week, a US aircraft shot down a Syrian government jet that had bombed USA -backed fighters fighting the Islamic State in the militants' de facto capital, Raqqa.More news: NATO chief: US allies to spend $12 billion more this year
"Certainly, we consider such threats to the legitimate leadership of the Syrian Arab Republic unacceptable".
The White House statement on Monday night was made without forewarning and caught State Department officials by surprise.
The White House threat essentially draws a "red line" on chemical weapons in much the same manner President Barack Obama did.
Less than an hour after Spicer issued the statement, Trump was back to tweeting about the 2016 campaign, denouncing investigations into potential collusion between Moscow and his campaign aides as a "Witch Hunt!"
On Tuesday, deputy White House spokesman Sarah Huckabee Sanders said "all relevant agencies. were involved in the process from the beginning".More news: Iraqi military says it has retaken two Mosul neighborhoods from Islamic State
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