The Napier MP was dismissed from cabinet on Tuesday after news outlet Stuff reported Mr Nash's leak, which took place in 2020.
Prime Minister Chris Hipkins said Mr Nash's actions were inexcusable.
"I expect ministers to uphold the highest ethical standards and his actions raise perceptions of influence which cannot stand," he said.
Mr Hipkins has ordered a review by the cabinet secretary into Mr Nash's communications with donors.
In the 2020 incident, Mr Nash emailed two businessmen - Troy Bowker and Greg Loveridge - describing a lost battle to widen COVID-19 relief for business.
Mr Hipkins also revealed that the email in question had passed through the prime minister's office, then occupied by Jacinda Ardern in 2021, as part of an official information act (OIA) request.
It did not get shown to Ms Ardern or her chief of staff as it was deemed unnecessary to make public under the scope of the OIA request.
Opposition leader Chris Luxon says that admission is evidence of a cover up.
"What kind of culture is that if you don't flag those issues higher up the chain?" he said.
"There is a real need to actually look at all correspondence since 2017, since (Mr Nash) became a minister.
"You want to be reassured that there is not a piece of that behaviour that goes on and on and on."
Mr Nash, a four-term MP was a senior cabinet minister who had already been sanctioned by Mr Hipkins for indiscretions.
Earlier this month, Mr Nash revealed he lobbied Police Commissioner Andy Coster to appeal a sentence he believed to be too light, while also attacking soft-on-crime judgments.
Mr Hipkins stood by his minister until Tuesday, sacking him within hours of learning of the breach.
On Wednesday afternoon, Mr Nash released a lengthy statement on his social media, apologising to the prime minister, cabinet and caucus colleagues, his staff, Labour volunteers, Napier residents and his family.
"I got this wrong. Too many times. Fair cop. The prime minister did what he needed to do. I accept this with humility," he said.
He acknowledged "playing outside the rules and that's unacceptable" but stressed no wrongdoing.
"Never ever for personal gain, or to benefit anyone other than the people of NZ and my Napier constituents, but that doesn't matter in the end. I let them down by not getting it right."
Mr Nash has ruled out resigning as an MP which would force a by-election in the swing seat of Napier within months of the general election due on October 14.
Mr Luxon said Mr Hipkins should sack Mr Nash from his partyroom, however the Labour leader said he wouldn't as no caucus rules had been breached.
"If he's lost confidence in him in cabinet, how has he not lost confidence with him being in caucus? How can he not be sure that other information hasn't leaked?" Mr Luxon asked.
Mr Nash's Napier seat is in the Hawke's Bay, which took the heaviest destruction from Cyclone Gabrielle last month.
The Opportunities Party Raf Manji said the case proved the case for an Australia-style Anti-Corruption Commission.
"Political donations and corporate lobbying in New Zealand is a murky and poorly regulated area of politics," Mr Manji said.
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