The Napier MP was dismissed from his cabinet position on Tuesday after news outlet Stuff broke the news of the 2020 indiscretion.
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.
The allegations give rise to the prospect of a by-election in the swing seat of Napier, however Mr Nash told Radio NZ he had no intention of resigning.
"There will be no by-election," he said.
Opposition leader Chris Luxon has seized on the indiscretion, saying a review into Mr Nash's behaviour during his ministerial career was needed to assess whether other information has leaked.
Mr Luxon said Mr Hipkins should force the embattled MP's exit from caucus as well, or the prime minister was guilty of "walking past the standard".
"This is a really serious and egregious contravention of the cabinet manual," he told broadcasters Three.
"A cabinet minister with confidential information which has been discussed in cabinet, who is leaking and sharing that information with his donors ... it's totally, utterly inexcusable.
"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?"
Under NZ's electoral law, a by-election can be avoided if an MP resigns their seat within six months of a general election, which is being held on October 14.
That would make the cut-off for a by-election-triggering departure on April 15, the day former prime minister Jacinda Ardern is resigning her Mt Albert seat.
Mr Luxon on Tuesday night called for a by-election but softened on that position on Wednesday.
"None of us want to go to a by-election. I'm sure it's not helpful for what the people of Napier need," he said.
"The by-election is the consequence of ... how you handle the standard of behaviour. You hold yourself accountable for standards and say that is unacceptable or you kinda don't."
Mr Nash's Napier seat is in the Hawke's Bay, which took the heaviest destruction from Cyclone Gabrielle last month.
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