PALM BEACH, Fla. -- The United States is vowing to keep up the pressure on Syria after the intense nighttime wave of missile strikes from U.S. ships, despite the prospect of escalating Russian ill will that could further inflame one of the world’s most vexing conflicts.

The Trump administration signaled Friday that the U.S. could follow this week’s missile attack with new sanctions against Syria, CBS News senior foreign affairs correspondent Margaret Brennan reports.

Standing firm, the Pentagon was even probing whether Russia itself was involved in the chemical weapons assault that compelled President Trump to action. The attack against a Syrian air base was the first U.S. assault against the government of President Bashar Assad.

Much of the international community rallied behind Mr. Trump’s decision to fire the cruise missiles in reaction to this week’s chemical weapons attack that killed dozens of men, women and children in Syria. But a spokesman for Russian President Vladimir Putin warned that the strikes dealt “a significant blow” to relations between Moscow and Washington.

At the United Nations, Russia’s deputy ambassador, Vladimir Safronkov, strongly criticized what he called the U.S. “flagrant violation of international law and an act of aggression” whose “consequences for regional and international security could be extremely serious.” He called the Assad government a main force against terrorism and said it deserved the presumption of innocence in the chemical weapons attack.

U.S. officials blame Moscow for propping up Assad.

“The world is waiting for the Russian government to act responsibly in Syria,” Nikki Haley, the U.S. ambassador to the U.N., said during an emergency Security Council session. “The world is waiting for Russia to reconsider its misplaced alliance with Bashar Assad.”

Haley said the U.S. was prepared to take further action in Syria but hoped it wouldn’t be necessary.

The official Saudi Press Agency reported that King Salman complimented Mr. Trump in a telephone conversation for his “courageous decision.”

Saudi Arabia, one of the most vehement opponents of Assad, said the missile barrage was the right response to “the crimes of this regime to its people in light of the failure of the international community to stop it.”

In Florida with the president, Treasury Secretary Steven Mnuchin said additional economic sanctions on Syria were being prepared.

Thursday night’s strikes -- some 60 cruise missiles fired from two ships in the Mediterranean -- were the culmination of a rapid, three-day transformation for Mr. Trump, who has long opposed deeper U.S. involvement in Syria’s civil war. Advisers said he was outraged by heartbreaking images of young children who were among the dozens killed in the chemical attack.

White House officials caution that Mr. Trump is not preparing to plunge the U.S. deeper into Syria. Spokesman Sean Spicer said the missile attack sent a clear message to Assad, but he avoided explicitly calling for the Syrian to leave office.

The impact of the strikes was also unclear. Despite intense international pressure, Assad has clung to power since a civil war broke out in his country six years ago, helped by financial and military support from both Russia and Iran. Russian military personnel and aircraft are embedded with Syria’s, and Iranian troops and paramilitary forces are also on the ground helping Assad fight the array of opposition groups hoping to topple him.

Mr. Trump spent Friday in Florida, in private meetings with Chinese President Xi Jinping. U.S. officials noted that the timing of the strike had the possible added benefit of signaling to China that Mr. Trump is willing to make good on his threat to act alone to stop North Korea’s nuclear pursuits if Beijing doesn’t exert more pressure on Pyongyang.