Iran's recent overtures signaling cooperation, though, likely stop short at the topics of Palestine and Syria. Iran is Syria's closest ally in the region.
In his speech Tuesday, Rouhani blasted what he described as the "structural violence" against Palestinians. While he never mentioned U.S. ally Israel by name, the intention of his comments was clear.
"Palestine is under occupation; the basic rights of the Palestinians are tragically violated, and they are deprived of the right of return and access to their homes, birthplace and homeland," he said. "Apartheid as a concept can hardly describe the crimes and the institutionalized aggression against the innocent Palestinian people."
On Syria, he said that the common objective of the international community regarding that country "should be a quick end of the killing of the innocent."
"We defend peace based on democracy and the ballot box everywhere, including in Syria, Bahrain and other countries in the region, and believe that there are no violent solutions to world crises," Rouhani said.
"While condemning any use of chemical weapons, we welcome Syria's acceptance of the Chemical Weapons Convention," he said, stressing that extremists' access to chemical weapons was the "greatest danger to the region."
Syria is under U.S.-led pressure to give up its chemical weapons arsenal in the aftermath of the August 21 attack on suburban Damascus that Washington and its allies blame on the al-Assad regime.
Obama said Tuesday that Syria's use of chemical weapons tested the relevance of the United Nations in the modern world, and he rejected contentions by the al-Assad regime and its main ally, Russia, that rebel forces were responsible for the attack.
"It is an insult to human reason -- and to the legitimacy of this institution -- to suggest that anyone other than the regime carried out this attack," Obama said.
Russia has blocked U.S. efforts to secure a strong Security Council resolution authorizing possible military force if Syria fails to comply with international regulations on turning over its chemical stockpiles. Obama argued Tuesday that such a resolution was vital.
"There must be a strong Security Council resolution to verify that the Assad regime is keeping its commitments, and there must be consequences if they fail to do so," he said. "If we cannot agree even on this, then it will show that the U.N. is incapable of enforcing the most basic of international laws. On the other hand, if we succeed, it will send a powerful message that the use of chemical weapons has no place in the 21st century, and that this body means what it says."
At the same time, Obama announced an additional $340 million in U.S. aid "to meet humanitarian needs in Syria and surrounding countries," referring to war refugees and other victims. The additional money increases the total U.S. commitment in humanitarian aid to $1.3 billion.
But Farah Atassi, activist with the opposition Syrian National Coalition, said Obama did not go far enough.
"The president did not address the core issue behind the Syrian crisis, which is holding the Assad regime accountable" -- including for its use of conventional weapons, she said in a CNN interview. More than 100,000 people have died in the conflict, the vast majority from conventional weapons, according to U.N. figures.
Obama is letting Syria's ally Russia lead, she said. "We don't trust the Russians."
"We want the U.S. to step forward and take the leadership right now to push for Assad to step down and allow for a transitional government from the opposition."
Emphasizing that the opposition has not asked for "boots on the ground," Atassi said U.S. military action should remain "on the table." And, she said, the rebels are looking to Washington for help empowering moderate elements of the opposition.
"We are not terrorists," she said.
Militant groups make up part of the Syrian opposition.
In his remarks to open the General Assembly on Tuesday, U.N. Secretary-General Ban Ki-moon called for the Syrian government to "fully and quickly" honor its obligations under the Chemical Weapons Convention, which calls for turning over control of its stockpiles.
Ban also appealed to all sides to stop supplying any weapons to all parties in the Syrian civil war while urging both the Syrian government and the opposition to respect international humanitarian law.
Russia and Iran, as well as Hezbollah in Lebanon, are providing military backing to the al-Assad regime while the United States and some European allies have started supplying light arms to rebel fighters.
Meanwhile, U.N. weapons inspectors will be back in Syria on Wednesday to assess at least six claims of chemical weapons use in Syria by the regime or rebels, a spokesman for Ban said Tuesday.
Al-Assad hinted at potential trouble for inspectors coming into Syria, saying other countries may order terrorists to attack them.
"Those militants might want to stop (the) experts' arrival. We know that those terrorists are under the control of some countries," he said in an interview Sunday with Chinese television. "And those countries may encourage the terrorists to stop experts from arrival, so that they could accuse the Syrian government for violating the agreement."