The army said rebels had launched a widespread attack on civilian areas and hit the hospital.
It accused groups including al-Nusra Front, Ahrar al-Sham and Jaish al-Islam of being behind the shelling.
SOHR also reported casualties at al-Dabbit Hospital, adding that the building had been heavily damaged.
It said that rebel shelling of government-held areas killed at least 19 people, including three children.
Rebel gains claimed
Zouhir Al Shimale, a local journalist, cast doubt on the veracity of the Syrian government's claims about the shelling of al-Dabbit Hospital.
"The hospital is 6km away from the rebel held area," he told Al Jazeera via the messenger service Whatsapp. "Rebels' guns or simple weapons couldn't have shelled the facility.
"Syrian state media is trying to put the blame on the Free Syrian Army (FSA) to deflect attention from Assad's campaign in Aleppo city."
| Several hospitals and clinics have previously been hit by deadly air strikes on rebel-held areas of Aleppo
On the other hand, Ahmad, a local FSA fighter who declined to provide his surname, claimed rebel groups were making gains against government forces in Muhanna and Halab Jadaydeh.
"The fighters have several positions from Assad's militias so far and the clashes are fierce and ongoing," he told Al Jazeera on Tuesday.
"This attack is a response to Assad's continuing campaign of air strikes on the city and his targeting and killing of civilians here."
Several hospitals and clinics have previously been hit by deadly air strikes on rebel-held areas of Aleppo.
'Crimes against humanity'
Tuesday's deaths come on top of the more than 250 people killed in less than two weeks across the city.
The Syrian opposition has accused the government of "war crimes and crimes against humanity" in Aleppo as barrel bombs - crude and unguided explosives - have been dropped on the city and surrounding areas.
Fayad Mohammad, a resident of al-Sheikh Fares district in eastern Aleppo, said his neighbourhood was struck seven times on Tuesday.
He said barrel bombs and shells had been pounding his neighbourhood.
| Government warplanes are targeting Aleppo's rebel-held areas [EPA/SANA]
"There is a lot of damage," he told Al Jazeera.
"We hoped that we would be included in yesterday's ceasefire agreement, but the warplanes are still in the sky.
"We haven't been able to go to our house since early morning because there is heavy artillery fire and shelling. We might die or get very badly injured by the random strikes."
Hamed Kasem, who lives in the Shaar area of eastern Aleppo, said he had seen many civilians injured as the attacks continued in his neighbourhood.
"Both today and last night, we haven't been able to rest because the sound of helicopters and bombardment."
Besides the dead, more than 1,500 people have been wounded in Aleppo as a result of the fighting.
In a field hospital, one of the injured, Abu Ali, told Al Jazeera's Amro Halain that he lost a leg and the use of an arm as a result of a missile strike.
"I was injured in Helanieh. I was hit when a missile struck our area. I have lost my leg and the use of my arm," he said.
"When people came to rescue us, another missile hit us. They were only two minutes apart."
Against this backdrop of the military onslaught, diplomatic efforts have intensified for Aleppo to be included in a ceasefire.
READ MORE: Syrian army violates Eastern Ghouta truce as rebels clash
De Mistura, the UN envoy, said on Tuesday that he wanted to focus on bringing a cessation of hostilities in Syria back on track at a meeting with Sergey Lavrov, Russia's foreign minister, in Moscow.
Lavrov said he expected to have a fruitful conversation, adding that US and Russian military are currently holding talks on the Aleppo ceasefire.
"I hope that in the coming hours such an agreement will be announced," Lavrov said after the meeting in Moscow.
De Mistura's meeting with Lavrov comes a day after he met John Kerry, the US secretary of state, in Geneva.
The US and Russia, a close ally of the Syrian government, are considered key to resolving the five-year war.