The Syrian army was meanwhile poised to launch an offensive against rebels who control part of the northern city.
Secretary of State John Kerry expressed "outrage" over Wednesday's air strike that hit Al-Quds hospital in Aleppo's rebel-held Sukkari neighbourhood.
He said it appeared to be "a deliberate strike on a known medical facility" and said Russia, which backs Syrian President Bashar al-Assad, must restrain the Damascus regime.
"Russia has an urgent responsibility to press the regime to fulfil its commitments under UNSCR 2254, including in particular to stop attacking civilians, medical facilities, and first responders, and to abide fully by the cessation of hostilities."
UN Secretary-General Ban Ki-moon said attacks that target civilians are "inexcusable" violations of humanitarian law.
"There must be accountability for these crimes," he said.
The city's last remaining paediatrician and three children were among the 20 people killed in the air strike overnight on the Al-Quds hospital, which was supported by Doctors Without Borders (MSF).
A civil defence group known as the White Helmets told AFP 30 people were killed in the strike on the hospital and a nearby block of flats.
- Truce 'barely alive' -
The Aleppo violence has raised fears for the ceasefire in other areas of Syria and called into question the future of UN-brokered peace talks in Geneva that have now gone into recess.
UN envoy Staffan de Mistura warned that the hard-won February 27 ceasefire was now "barely alive" and could collapse "any time".
Speaking late Wednesday after a third round of Syria peace talks fizzled out, De Mistura also urged its co-sponsors Russia and the United States to take action to rescue it.
He said the United States, which supports some rebel groups, and regime ally Russia needed to act, calling on them to organise a high-level Syria meeting before negotiations resume.
UN aid chief Stephen O'Brien further raised the alarm on Thursday in remarks made at the UN Security Council.
"We must all be ashamed this is happening on our watch," said O'Brien, urging world powers to salvage the truce.
"You must not squander the opportunity presented by talks in Geneva and by the cessation of hostilities to put an end to the massive human suffering in Syria," he said.
And the UN's Jan Egeland, who heads an international humanitarian taskforce for war-ravaged Syria, spoke of a "catastrophic deterioration" of the situation in the country.
"The stakes are so incredibly high," Egeland told reporters in Geneva, and warned that "the lifeline to hundreds of thousands, if not millions of people... may be broken."
- 'Humanitarian disaster' -
Rebel rocket and artillery fire on government-held neighbourhoods on Thursday killed 22 civilians, including two children, the Syrian Observatory for Human Rights said.
Regime air strikes on rebel-held districts, including Bustan al-Qasr, killed 31 civilians, including three children, said the British-based monitor.
An AFP correspondent said every building in sight in the Bustan al-Qasr district had had its windows blown out.
"It is the worst day in Aleppo in five years. The regime did not spare a single neighbourhood," said one resident.
The International Committee of the Red Cross warned that Aleppo city was "on the brink of humanitarian disaster".
The pro-government Al-Watan newspaper and a Syrian regime source meanwhile said the army was preparing to take the whole of Aleppo.
"Now is the time to launch the battle for the complete liberation of Aleppo," the paper said, adding that it "will not take long to begin, nor to finish".
A regime source told AFP that "the army is preparing a huge operation in the coming days to push the rebels away from the city by encircling it and creating a security zone".
Rebels have controlled eastern districts of Aleppo city since 2012, while western neighbourhoods are held by the regime.
Control of the surrounding province is divided between a myriad of armed groups -- jihadists of Al-Qaeda and the Islamic State group, Kurdish militia and various rebel factions as well as the army.
Further north in the province, rebels including the powerful Islamist Ahrar al-Sham group and Kurdish forces fought fierce battles Wednesday and early Thursday that left 64 fighters dead, a monitor said.
More than 270,000 people have been killed in Syria and millions forced from their homes since the conflict erupted in 2011.