CNN cancels interview with Iranian president over headscarf demand
Amanpour pointed out that no previous presidents had requested this when she interviewed them outside Iran.
She says a Raisi aide told her it was because of "the situation in Iran", Qazet.az reports.
The death of a woman detained in Iran for allegedly breaking headscarf rules has sparked violent unrest there.
Mahsa Amini, 22, fell into a coma last week, hours after morality police arrested her.
Officers reportedly beat Ms Amini's head with a baton and banged her head against one of their vehicles. The police have said there is no evidence of any mistreatment and that she suffered "sudden heart failure".
The protests, now in their seventh day, have reached 80 other cities and towns in the Islamic Republic. At least 17 people have been killed.
The interview would have been Mr Raisi's first on US soil, during his visit to the UN General Assembly.
Amanpour said she had been ready to conduct it when one of the president's aides insisted she covered her hair at Mr Raisi's request.
"We are in New York, where there is no law or tradition regarding headscarves," she said later on Twitter.
Amanpour said Mr Raisi's aide had made it clear an interview would not happen if she did not wear a headscarf, saying it was "a matter of respect".
Her team walked away from the interview refusing what she called the "unprecedented and unexpected condition".
The US anchor later posted a picture of herself without a headscarf in front of an empty chair where Mr Raisi would have sat for their interview.
Mr Raisi was elected last year and signed an order in August to enforce a new list of restrictions.
These included the introduction of surveillance cameras to monitor and fine unveiled women or refer them for "counselling", and a mandatory prison sentence for any Iranian who questioned or posted content against the hijab (headscarf) rules online.
The restrictions led to an increase in arrests but also sparked a surge in women posting photos and videos of themselves without headscarves on social media - something that has only intensified in the days following Ms Amini's death.