Human rights organizations have chronicled Iran's treatment of some religious minorities, particularly members of the Baha'i faith.
Despite the recent release of more than a dozen political and religious prisoners in Iran, "religious minorities and other prisoners of conscience in Iran continue to suffer for their beliefs, including more than 100 imprisoned members of the Baha'i community and its leadership," said Robert P. George, chairman of the U.S. Commission on International Religious Freedom, in a statement Monday.
Also, "a renewed crackdown on Protestant Christians in recent weeks has led to numerous arrests," the commission said. "Of those Christians already imprisoned, Iranian-American pastor Saeed Abedini continues to languish in Evin prison while his eight-year sentence was upheld just last month."
Rouhani told CNN he is pursuing a citizenship charter that will work to improve freedoms. "So I will spare no effort to ensure that those who are currently in prison will see an opening door," he said.
Iranian president on Syrian conflict
Rouhani also addressed Iran's role in the Syrian civil war.
Some Iran troops are fighting on the side of President Bashar al-Assad's regime, Amanpour noted.
Rouhani insisted the Iran's involvement does not even number hundreds of people.
"We have close relations with Syria from a long time ago," he said, adding that Iran has military attaches and experts stationed there.
Asked about Iranian weaponry being used by the Syrian regime, Rouhani cautioned against baseless "propaganda."
"Are you encouraging the regime to give up its chemical weapons as the deal between the U.S. and Russia says?" Amanpour asked.
"We believe in general that the entire region of the Middle East has -- as far as that region is concerned -- all weapons of mass destruction, including chemical weapons, have to be eradicated from the region," Rouhani responded.
Iran is glad Syria has committed itself to the Chemical Weapons Convention, he said.
Rouhani on why no meeting with Obama
"I would like to say to American people: I bring peace and friendship from Iranians to Americans," Rouhani said in English, marking the first time he's spoken the language in a TV interview since becoming president. For the rest of the interview Rouhani spoke Farsi.
In many ways, Rouhani is the "it" man of the U.N. General Assembly, as Western leaders look to gauge whether his diplomatic overtures will translate into concrete policy changes.
There was widespread speculation that he and U.S. President Barack Obama might meet face-to-face.
"There were some talks about" a possible meeting, Rouhani told Amanpour through a translator. "And preparation for the work was done a bit as well."
But no such meeting happened.
Two senior U.S. administration officials told CNN Tuesday that the encounter was called off because it was considered "too complicated" for Iran back home.