President Obama, speaking to world leaders at the U.N. General Assembly Tuesday, vowed the U.S. "will do what we must" to prevent Iran from gaining a nuclear weapon -- calling this a threat to Israel's existence.

Amid accusations from Republicans, including Mitt Romney, that Obama's policies have not slowed Iran's nuclear march, the president used the U.N. stage to assure the international community that he is serious about preventing that outcome.

"Make no mistake: a nuclear-armed Iran is not a challenge that can be contained," Obama said. "It would threaten the elimination of Israel, the security of Gulf nations, and the stability of the global economy. It risks triggering a nuclear-arms race in the region, and the unraveling of the non-proliferation treaty. That's why a coalition of countries is holding the Iranian government accountable. And that's why the United States will do what we must to prevent Iran from obtaining a nuclear weapon."

The president reiterated that he wants to resolve the issue "through diplomacy" but the time for doing so "is not unlimited."

Obama has been under fire for, so far, opting not to meet in New York with visiting Israeli Prime Minister Benjamin Netanyahu, whose prime area of international concern is Iran's nuclear program. Obama also has not scheduled one-on-one meetings with any world leader at the U.N.

The reference to Iran came toward the end of a speech otherwise devoted to addressing the recent tumult in the Middle East and North Africa, including the murder of the U.S. ambassador to Libya and three other Americans in Benghazi.

Obama began his address to the General Assembly Tuesday with a tribute to Ambassador Chris Stevens. Recalling Stevens' time serving in the Peace Corps as an English instructor in Morocco, he said Stevens "came to love and respect" the people of the region and carried that commitment throughout his life.

"I tell you this story because Chris Stevens embodied the best of America," Obama said.

The president went on to restate his administration's support for the Arab Spring, calling it a "season of progress."

But he said the recent violence and unrest is indicative of the difficulties along the way. "True democracy -- real freedom -- is hard work," he said.

Obama said leaders in the region are at a critical juncture, and urged them to choose the forces of hope over the forces of intolerance.

"It is time to leave the call of violence and the politics of division behind," Obama said. "On so many issues, we face a choice between the promise of the future, or the prisons of the past. And we cannot afford to get it wrong. We must seize this moment. And America stands ready to work with all who are willing to embrace a better future."

The president called on world leaders to "marginalize" those that stoke hatred of the West in order to further their own politics.

And he continued to address the anti-Islam film that is blamed for many of the recent demonstrations against U.S. diplomatic posts -- though that film may not have played much of a role in the attack on the U.S. Consulate in Libya. While the president did not describe that attack as terrorism Tuesday, Romney did in a separate speech to the Clinton Global Initiative conference.

Obama stressed that while he condemns the "crude and disgusting" video, America maintains the right to free speech.

"And on this we must agree: There is no speech that justifies mindless violence," Obama said.

"There are no words that excuse the killing of innocents. There is no video that justifies an attack on an embassy," the president said.

Obama said the world now faces a choice "between the forces that would drive us apart, and the hopes that we hold in common."

The last time Obama addressed the assembly, there was an air of hope surrounding the Arab Spring. U.S. officials remain optimistic, but some also worry that the latest unrest is perhaps the dark side of the revolution.

Obama used his U.N. address to urge leaders in the region not to let their hard-fought gains be undermined by those peddling the politics of hate and division.

The relationship between the Libya attack and the protests against an anti-Islam film elsewhere in the region remains unclear. Obama, in an interview on Monday, said the Libya attack was not just a "mob action." Other evidence has emerged indicating the attack was pre-planned, though the administration has not yet publicly drawn that conclusion.

The speech Tuesday morning kicked off a day heavy on foreign policy for both the president and Romney.

Romney addressed the Clinton Global Initiative in New York City in the morning, and Obama is expected to address the initiative later in the day.