DAMASCUS, Syria — Insurgent bombs exploded at a Damascus school building on Tuesday, and Syrian government shells landed in the disputed Golan Heights region held by Israel, underscoring the Syrian conflict’s growing scope as it dominated much of the discussion on the first formal day of the United Nations General Assembly.

President François Hollande of France called for outside military intervention to protect areas in Syria held by the rebels and told the General Assembly in his first speech there that the government of President Bashar al-Assad, which has been widely accused by Western and Arab countries of brutality in its campaign to crush the insurgency, “has no future among us.”

President Obama, in his remarks to the General Assembly, said Syria’s future “must not belong to a dictator who massacres his people.” But neither he nor Secretary of State Hillary Rodham Clinton detailed new initiatives to address the spiraling violence in Syria, where an estimated 20,000 people have been killed and millions have been displaced since the anti-Assad uprising began with peaceful protests in March 2011.

“If there is a cause that cries out for protest in the world today, it is a regime that tortures children and shoots rockets at apartment buildings,” Mr. Obama said, “and we must remain engaged to assure that what began with citizens demanding their rights does not end in a cycle of sectarian violence.”