Speaking from the podium of the General Assembly Hall, Clinton accused Iran of "flouting the rules" of the Nuclear Nonproliferation Treaty with its suspect uranium enrichment program, and said it is "time for a strong international response."
For his part, Ahmadinejad earlier rejected such allegations, saying Washington has offered not "a single credible proof."
They were the opening salvos in four weeks of deliberations over how to improve the NPT, formally reviewed every five years in a meeting of all 189 treaty members - all the world's nations except India, Pakistan, Israel and North Korea.
The review conference is meant to produce a final document pointing toward ways to better achieve the NPT's goals of checking the spread of nuclear weapons, while working toward reducing and eventually eliminating them.
Because it requires a consensus of all parties, including Iran, any formal final decision would be highly unlikely to censure the Tehran government, which would block consensus.
Instead, as delegates assess the state of the NPT in U.N. conference halls, U.S. and European diplomats will be working elsewhere to reach agreement with sometimes reluctant China and Russia on a fourth round of U.N. Security Council economic penalties to impose on Iran.
"I hope that we can reach agreement in the Security Council on tough new sanctions," Clinton told reporters, "because I believe that is the only way to catch Iran's attention."
In her address, Clinton proposed that the nonproliferation treaty be strengthened by introducing "automatic penalties" for noncompliance, rather than depend on such drawn-out council negotiations.
Ahmadinejad devoted much of his half-hour speech to the huge U.S. nuclear arsenal, denouncing the Obama administration's refusal to rule out the use of those weapons.
"Regrettably, the government of the United States has not only used nuclear weapons, but also continues to threaten to use such weapons against other countries, including Iran," Ahmadinejad said.
He referred to the new U.S. Nuclear Posture Review's provision retaining an option to use U.S. atomic arms against countries not in compliance with the nonproliferation pact, a charge Washington lays against Iran.
Clinton later announced the U.S. government was releasing previously undisclosed details about the U.S. arsenal. About the same time in Washington, the Pentagon was reporting that the U.S. maintains 5,113 nuclear warheads in its stockpile and "several thousand" more retired warheads that await dismantling.
Ahmadinejad invited President Barack Obama to join a "humane movement" that would set a timetable for abolishing those and all other atomic arms, weapons he called "disgusting and shameful."
As the Iranian president spoke, the U.S. delegation, of working-level staff, walked out of the General Assembly hall, joined by several European delegations, including the French and British. Lower-level Iranian officials sat through Clinton's later speech.
Yukiya Amano, head of the U.N. watchdog International Atomic Energy Agency, leveled the specific indictment against Iran, saying his inspectors could not confirm that all of its nuclear material is devoted to peaceful activities.
Iran must "clarify activities with a possible military dimension," Amano said.
Opening the conference, U.N. Secretary-General Ban Ki-moon said that "the onus is on Iran" to clear up doubts. He said the same in a later private meeting with Ahmadinejad, the U.N. chief's office said.
Ahmadinejad, the only head of state participating in the conference, complained that the U.S. and its allies were pressuring Iran "on the false pretext of probable diversions in their peaceful nuclear activities without providing even a single credible proof to substantiate their allegation."
The Iranian leader reiterated his country's support for establishing a nuclear weapons-free zone in the Middle East, an Arab-backed idea aimed at Israel's unacknowledged nuclear arsenal of perhaps 80 bombs.
Clinton, too, repeated U.S. endorsement of the idea. She told reporters regional instabilities mean conditions aren't right at this time for a Mideast "nuke-free" zone, "but we are prepared to support practical measures for moving toward that objective."
Egypt has proposed that this 2010 NPT conference back a plan calling for the start of negotiations next year on such a Mideast zone. The proposal may become a major debating point in the monthlong session.