The G7 group of nations has failed to reach agreement over threatening new sanctions against Russia and Syria.
Foreign ministers were seeking a common position on the Syrian conflict, before the US secretary of state flies to Russia to try to persuade it to abandon its Syrian ally.
The nations agreed there was no solution to the Syria crisis with President Assad in power.
But UK proposals to target sanctions at senior military leaders were sidelined.
The diplomacy in the Italian town of Lucca follows the latest apparent use of chemical weapons in Syria.
Syria has denied it carried out a chemical attack on the rebel-held town of Khan Sheikhoun last week that left 89 people dead.
In response, the US fired 59 cruise missiles at a Syrian airbase that it said was implicated in the attack.
The Syrian military says the damage to the base was less than claimed by the US
Speaking after the end of the G7 meeting, US Secretary of State Rex Tillerson said the missile strike “was necessary as a matter of US national security interest”.
“We do not want the regime’s uncontrolled stockpile of chemical weapons to fall into the hands of Isis [so-called Islamic State] or other terrorist groups who could and want to attack the United States or our allies.”
He also added that the US saw “no further role for the Assad regime longer term, given that they have effectively given up their legitimacy with these kinds of attacks”.
Mr Tillerson will head to Moscow for talks on Syria later on Tuesday, hoping to persuade the Russians that they have an unreliable ally in President Assad.
Italian foreign minister Angelino Alfano – hosting the G7 talks – said ministers wanted to engage with Russia to put pressure on President Assad, adding that “we must not push Russia into a corner”.
“We think the Russians have the leverage that is needed to put pressure on Assad and to get him to observe the commitments with regard to the ceasefire,” he added.
What will US achieve in Moscow – BBC’s Steve Rosenberg
The fact that Rex Tillerson’s visit to Moscow is happening at all is telling.
Russia reacted angrily to last week’s US missile strike on Syria, condemning it as an “act of aggression”. Yet Moscow is happy to host the US secretary of state. He’ll meet his Russian counterpart Sergei Lavrov and a meeting with President Putin cannot be ruled out.
But experience shows that Moscow does not take well to threats or ultimatums.
If Mr Tillerson thinks he can weaken Moscow’s support for President Assad, he may need to re-think. The Syrian president is Russia’s key military ally in the Middle East. Russia has invested heavily – militarily, politically and financially – to keep him in power.
New details on chemical attack and retaliation
Reports on Monday quoted a senior US official as saying that the Russians knew of the chemical attack because a drone had been flying over a hospital in Khan Sheikhoun as victims sought help.
Hours later a jet bombed the hospital in what the US believed was an attempt to cover up the attack, the Associated Press agency said.
Meanwhile, US Defence Secretary James Mattis gave fresh details on the retaliatory strike against Syria’s Shayrat airbase.
He said the “measured response” by the US had “resulted in the damage or destruction of fuel and ammunition sites, air defence capabilities and 20% of Syria’s operational aircraft”.
The Syrian military admits significant material damage but a Russian defence ministry spokesman said only six Syrian Air Force MiG-23s, plus a number of buildings, were destroyed and that only 23 of the missiles had reached Shayrat.
What are Syria and its allies saying?
Syria has denied using any chemical agents, and Russia says the US has failed to provide evidence Syria has chemical weapons.
Russia and Iran, President Assad’s key military backers, are also threatening retaliation if there are any further American air strikes, saying the US attack had crossed “red lines”.
“From now on we will respond with force to any aggressor or any breach of red lines from whoever it is and America knows our ability to respond well,” said a statement from a joint command centre comprising the forces of President Assad’s allies.
What is the US policy on Syria?
In recent days there have been mixed messages from the US on its priorities in Syria.
Mr Tillerson said on Sunday that there had been “no change to our military posture” in Syria following the US airbase strike and that Washington’s “first priority” was to defeat so-called Islamic State (IS).
Those comments came a day after the US’s ambassador to the United Nations, Nikki Haley spoke about the removal of President Assad, saying: “In no way do we see peace in that area with Assad as the head of the Syrian government.”
There was further confusion on Monday. Mr Tillerson spoke in public about an interventionist approach, saying: “We rededicate ourselves to holding to account any and all who commit crimes against the innocents anywhere in the world.”
Hours later, White House press secretary Sean Spicer suggested Mr Trump would act against Syria not just if it used chemical munitions.
“If you gas a baby, if you put a barrel bomb into innocent people, I think you will see a response from this president,” Mr Spicer said in his daily briefing. The White House later said Mr Spicer had meant to refer to barrel bombs containing “industrial chemicals”.