Both the government and the museum's managers have denied responsibility.
Italian Culture Minister Dario Franceschini, who accompanied Rouhani on the museum trip, called the cover-up "incomprehensible", insisting that neither him nor premier Matteo Renzi had been made aware of the decision in advance.
Meanwhile, a smiling Rouhani told reporters he had "no contact on the subject" with Italian authorities.
"I know that the Italians are very hospitable, a people who seek to make their guests' visits as pleasant as possible and I thank them for that," he added.
Rouhani and Renzi made speeches in Rome's Capitoline Museums on Tuesday, with a huge statue of Roman emperor Marcus Aurelius on a horse featuring prominently in many of the photographs of the event.
But nude statues, including a Venus dating from the second century BC, had all been covered up in temporary wooden cartons, removing the risk of them creeping into any of the shots - or catching Rouhani's eye.
Sources at Rome's City Hall, which manages the venue, assured the press that the municipality had played no role in the decision, palming journalists off to the premier's office.
Franceschini then attempted to calm things down, telling Corriere: “I never spoke about it being the responsibility of the Capitoline superintendents. It's clear that there was an excess of zeal on the part of those in charge of organizing such events, who made the decision, without telling, as I already said, either the premier or I.”
Paolo Aquilanti, the general secretary of the premier's office, has reportedly ordered an inquiry.
But Renato Brunetta, the lower house whip for Silvio Berlusconi's Forza Italia party, took to twitter in defense of the museum's management, alleging the request came from the premier.
Sovrintendenza capitolina sbugiarda @matteorenzi. Statue coperte per volere Palazzo Chigi. Figuraccia mondiale. Vergogna- Renato Brunetta (@renatobrunetta) January 27, 2016
Codacons, the Italian consumers' association, also waded in, saying the story, which was picked up by major international news outlets, has tarnished Italy's reputation, while demanding that whoever was responsible – if the culprit is ever exposed – be fired.
They should be punished “for the severe damage done to the honour and image of Rome, and the whole of Italy, and for the disgrace inflicted on the country globally,” the association added.
Alessandro Di Battista, from the Five Star Movement party, said the story was akin to the shame of the ostentatious funeral of a mafia boss held in Rome last August.
“There's government outrage but zero liability, in the end the custodian will pay.”
Corriere journalist Pierluigi Battista wrote in an editorial: “It's a stain. We have nothing to be ashamed of. We mustn't think that nude art is something shameful or contempitible."
Staff at another museum were equally outraged.
“It was a crazy thing to do,” Paolo Severino, who works at a museum in central Rome, told The Local.
“Nude art is everywhere in Italy. And anyone who comes to a museum here knows what to expect. I've never seen or heard about anyone getting offended by it.”