"There must be accountability for these crimes," Ban said in a statement yesterday.
The city's last remaining pediatrician 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).
Ban called on the warring sides to "immediately renew their commitment" to a ceasefire that had been in force since late February.
He encouraged the international powers backing Syria's peace process, in particular the United States and Russia, to "ensure credible investigations of incidents such as the attack on Al-Quds hospital."
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UN aid chief Stephen O'Brien separately appealed to world powers to revive the ceasefire and put an end to the "massive human suffering" in Syria.
"We must all be ashamed this is happening on our watch," O'Brien told the Security Council during a meeting on the appalling humanitarian crisis stemming from the five-year war.
Five Security Council members -- Egypt, Japan, New Zealand, Spain and Uruguay -- are drafting a new resolution condemning attacks on hospitals in war zones such as Syria, but also Yemen, Afghanistan and South Sudan.
"We are trying to shine a spotlight on this issue because it's clearly a fundamental problem in the way in which modern warfare is being conducted in these horrible situations," said Bohemen.
More than 270,000 people have been killed in Syria and millions forced from their homes since the conflict erupted in 2011.