A similar address by Iran's Supreme Leader Ali Khamenei made a similar point.
"We will not sign any deal unless all sanctions are lifted on the same day ... We want a win-win deal for all parties involved in the nuclear talks," Rouhani said.
Khamenei called the deal "non-binding" and said the prospect of lifting sanctions in stages was "unacceptable", saying they must be removed on the same day a deal is signed.
He also said that an extension of the June 30 deadline "would not be the end of the world" and that the talks between Iran and the powers were only about the nuclear issue.
The United States has made clear that it would only agree to sanctions on Iran being phased out gradually as Tehran complies with a final deal to curb its nuclear program.
"You can't start talking about relieving sanctions until we've reached agreements about how we're going to shut down every pathway they have to a nuclear weapon," White House spokesman Josh Earnest said at a news briefing.
"It has never been our position that all of the sanctions against Iran should be removed from Day One," he said.
The United States could foresee a point at which sanctions could be dismantled, but only after "sustained compliance over a long period of time," Earnest said.
US President Barack Obama defended the framework agreement on Tuesday. He told NPR that Iran would be kept a year away from obtaining a nuclear weapon for more than a decade, but conceded that the buffer period could shrink to almost nothing after 13 or more years.
Prime Minister Benjamin Netanyahu said Wednesday that he agreed with the president's statement that Iran would have a near zero breakout time – but unlike Obama, Netanyahu said that the minimal breakout time to create the bomb would happen from the moment of the deal's inception.
According to Netanyahu the deal would be "the inevitable result of the automatic removal of restrictions, something that would enable Iran to gain an industrial-scale production capacity."
A deadline for reaching the final agreement is June 30 and Iran, the United States, Germany, France, Britain, Russia and China are expected to meet in the coming days to discuss the deal.
Associated Press, Atilla Somfalvi, and Moran Azulay contributed to this story.