Kerry said he was ready to meet the Russians again on Friday to see if there are any means to revive the truce that failed this week, but diplomats were pessimistic.
"The only way to achieve that is if the ones that have the air power in that part of the conflict simply stop using it," Kerry told reporters after the talks.
"Not for one day or two, but for as long as possible so that everyone sees that they are serious."
UN peace envoy Staffan de Mistura called it a "long, painful and disappointing meeting" but insisted that Washington and Moscow are serious about the truce.
He blamed unnamed other parties among the delegates for "undermining" the US-Russian initiative and added "they are still trying, so declaring it dead would be wrong."
But he had a less rosy view of events on the ground.
"Meanwhile, what is happening in Aleppo is under attack and everyone is going back to the conflict," he said.
"The next few hours -- days at maximum -- are crucial for making it or breaking it."
In Damascus, the Syrian army urged residents of Aleppo to stay away "from the positions of terrorists" as it launched its new offensive in defiance of the truce.
London-based watchdog the Syrian Observatory for Human Rights described the latest assault as "a large-scale land offensive supported by Russian air strikes."
- Rebel neighborhoods -
An AFP correspondent in the rebel-held east of Aleppo witnessed a dozen families fleeing the Soukkari district for other rebel areas further north.
Rebel-held neighborhoods of Aleppo had already suffered large-scale fires after a night of bombardment from what activists called phosphorous bombs.
The estimated 250,000 residents of east Aleppo, which rebels against Bashar al-Assad's have held since 2012, have been living under siege since early September.
The conflict in Syria has cost more than 300,000 lives since 2011, during which time more than half the population was uprooted from their homes.
On September 9, Kerry and Lavrov met in Geneva and agreed to call a ceasefire, with Moscow responsible for forcing Assad's forces to stand down and allow in UN aid convoys.
The United States was to pressure opposition rebel forces to obey the truce but both sides cried foul and on Monday this week the Syrian army declared the ceasefire over.
Diplomats believe the US-Russian Geneva process is the only available hope to end the five-year conflict, but Moscow and Washington have fallen out spectacularly.
- Phosphorous bombs -
Russia says Kerry failed to deliver a rebel ceasefire, and was furious when US-led coalition warplanes bombed a Syrian base, a strike Washington says was an error.
Washington in turn accused Russian jets of carrying out a deadly strike on a UN aid convoy on Monday, and Kerry and Lavrov exchanged angry words at the Security Council.
"I listened to my colleague from Russia and felt a little bit like we are in a parallel universe," Kerry declared on Wednesday after Lavrov tried to blame the rebels.
Kerry told the Council the only hope of reviving the ceasefire would be for Russia to order Assad to ground his air force and stop hitting civilian targets.
Food aid promised for Aleppo under the US-Russia deal has been stalled at the border since last week and will go bad in just a few days.
"Forty trucks are sitting at the Turkish-Syrian border. The food will be expiring on Monday," the head of the UN humanitarian taskforce for Syria, Jan Egeland, said.
"The drivers are sleeping at the border and they have done so now for now a week, so please, President Assad, do your bit to enable us to get to eastern Aleppo."
The UN resumed aid deliveries on Thursday after a pause in the wake of a strike on the convoy in Syria's north that killed 20 civilians and destroyed 18 aid trucks.
- Talks in 'next few weeks'? -
Nevertheless, the United Nations' deputy envoy for Syria said Thursday he hoped talks could resume in the coming weeks, despite "grim" events on the ground.
Ramzy Ezzeldin Ramzy said De Mistura is in talks with the parties to organize "direct negotiations," a departure from past rounds where the sides met with mediators.