Thousands of civilians poured out of Eastern Ghouta as the capture of a key town brought Syria's government even closer to retaking the devastated rebel enclave outside Damascus.

"The Russians made a deliberate choice not to restrain the Assad regime. Thus, the carnage in East Ghouta continues. Russia is morally complicit and responsible for Assad's atrocities," Pentagon spokeswoman Dana White told reporters.

"We urge Russia to compel the Assad regime to stop killing innocent Syrians and allow much needed aid to reach the people of East Ghouta and other remote areas. We support our diplomats in their efforts to resolve the conflict with the United Nations-backed Geneva process."

The 15 UN Security Council members -- including the United States -- unanimously adopted a resolution last month demanding a month-long ceasefire. But it has not been respected.

The Russia-backed Syrian army and allied militia began a fierce bombing campaign on February 18 on Eastern Ghouta -- the last opposition bastion near Damascus -- then launched a ground offensive a week later.

Russia, which was among those voting in favor of the UN text, indicated Thursday it would carry on backing Damascus in its military offensive.

"We will continue fighting terrorists, we will finish them off, we will help finish them off in Eastern Ghouta, where the Syrian army is now conducting operations with our support," Russian Foreign Minister Sergei Lavrov said.

The assault on Ghouta has left nearly 1,250 civilians dead, around a fifth of them children.

- 'Enabling atrocities' -

"Russia is either incompetent, committing illegal acts or both. Russia is propping up the Assad regime, not targeting Al-Qaeda affiliated terrorists in Syria. Russia could stop the civilian casualties in Syria," said White.

President Donald Trump's National Security Advisor, HR McMaster, said separately that "all civilized nations must hold Iran and Russia accountable for their role in enabling atrocities and perpetuating human suffering in Syria."

Speaking at the Holocaust Memorial Museum in Washington, McMaster said all nations must respond "more forcibly than simply issuing strong statements.

"It is time to impose serious political and economic consequences on Moscow and Tehran," he said without giving more detail, as Washington faces criticism for not following up its words with action.

Along with Moscow, Tehran is the Syrian regime's other major ally.

"Assad should not have impunity for his crimes and neither should his sponsors," McMaster said, adding that Moscow "has done nothing" to encourage Assad to ensure delivery of humanitarian aid, or to respect ceasefire and de-escalation agreements.

On Monday Washington's United Nations Ambassador Nikki Haley told the Security Council the US "remains prepared to act if we must."

She recalled -- as did McMaster on Thursday -- that Trump had ordered missile strikes on a Syrian air base in April last year in retaliation for a sarin gas attack blamed on Assad's forces.