More than 20 children were among the dead in the attack on the school compound in the rebel-held village of Hass in Idlib province Wednesday.
UN Secretary General Ban Ki-moon said he was "appalled" at repeated bombings of the school compound.
"If deliberate, this attack may amount to a war crime," he said in a statement.
Idlib, a main opposition stronghold is regularly hit by both Syrian and Russian airstrikes, as well as the US-led collation hitting ISIS targets.
The US, France and international monitors have said either the Syrian regime or its Russian backers carried out Wednesday's attacks, but Moscow has denied flying in the area.
Russian President Vladimir Putin blamed President Barack Obama Thursday for failing to reach an agreement to end the bloodshed, saying the Americans are "shooting themselves in the foot."
"Our personal agreements with the US president have not worked," Putin said, referring to aborted efforts to negotiate a cease-fire in Syria, which would have included some humanitarian relief.
"There were forces in Washington that did everything possible to fail to implement these agreements," Putin continued. "This shows an inexplicable -- I would say an irrational -- desire of Western countries to repeat their mistakes over and over again."
Russia and Syria claim to have implemented a "humanitarian pause" in rebel-held areas in Eastern Syria, including Aleppo, where UN officials are trying to evacuate hundreds of wounded and attempt to deliver medical and food aid into besieged areas.
"What Russia really wants from the UN is credit," US Ambassador to the United Nations Samantha Power said Wednesday in a blistering attack on Russia before the UN Security Council.
But she told them, "You don't get congratulations and get credit for not committing war crimes for a day, or a week."
UNICEF executive director Anthony Lake said Wednesday's attack "may be the deadliest attack on a school since the war began more than five years ago."
The Obama administration has not made a formal determination that any war crimes have been committed in Syria. But earlier this month Secretary of State John Kerry said Russia and the regime's targeting of civilians and medical facilities "beg for an appropriate investigation of war crimes."
He did not say what type of investigation he wanted to see or who would lead it.
"He recognizes that 'war crimes' has a very legalistic definition, and that, and that's why he wants it to be investigated," State Department spokesman John Kirby said Thursday. "He wants it to be looked at, he wants there to be a determination by the international community one way or another."
Kirby added that discussions are taking place among the international community about "next steps."
Last month, Ban called on the UN Security Council to refer the situation in Syria to the International Criminal Court, which many countries have endorsed. French Foreign Minister Jean-Marc Ayrault and British Foreign Secretary Boris Johnson have also said Russian and Syrian actions may constitute war crimes.
International rights groups and others have documented Russian and Syrian crimes, including alleged attacks on civilians and medical facilities, starvation as a tool of war, torture and use of chemical weapons on the battlefield.
But neither Syria nor Russia has ratified the ICC's founding Rome Statue, limiting the court's power without Security Council action. Russia would undoubtedly veto any action by that body.