The Home Secretary was sacked by Liz Truss over the breach, which she claimed to have “rapidly reported” upon becoming aware of it.
However, it has now emerged Ms Braverman emailed a member of staff telling them to "delete and ignore" the sensitive message that forced her resignation.
After sending the original email to the wrong person at 7.52am, she was told by the recipient it had been sent to them in error 38 minutes later.
The reply from Ms Braverman’s account was then sent at 10.02am and said: "Please can you delete the message and ignore."
Now Mr Gove has rejected calls for the Government to publish detailed documents on the breach, insisting he is “more than satisfied” with her reappointment as Home Secretary.
Labour is demanding the Government publish its assessments of the incident, something dismissed by the Levelling Up Secretary.
He told the BBC’s Sunday With Laura Kuenssberg programme: “When we publish everything, we also potentially publish information that can compromise the effective operation, not just of government, but of national security itself.
“I also, critically, want to ensure that what we don’t do is, on the basis of the imperfect information that is in the public domain, rush to judgement in a way that would seem to me to be inappropriate.”
Asked if Ms Braverman is a politician of integrity, Mr Gove said: “Absolutely.
“I am satisfied, more than satisfied, that in resigning, accepting responsibility, apologising, and then in being assured by the Cabinet Secretary and the Prime Minister that Suella coming back into office was the right thing, that Suella is now in a position to do the work that she is dedicated to doing.”
Confronted with the email, Mr Gove insisted her request was “standard practice”.
He suggested Ms Braverman is facing opposition because she is “brave” and “making changes”.
“You only take flak if you’re over the target,” the senior Tory said.
The former journalist also appeared to blame the media for the furore around the matter, saying: “It becomes a distraction if people are asking these questions.”
Mr Sunak is now under growing pressure over the appointment, with the SNP saying he had “engulfed” the party in scandal.
SNP Shadow Home Secretary Stuart McDonald MP said: "Rishi Sunak has blown his credibility, and broken his promise of integrity, by doing a dodgy backroom deal with Suella Braverman to shore up his own position.
"The Tory Home Secretary's appalling record of breaking the ministerial code, and repeatedly leaking sensitive government information, makes her singularly unfit for office.
"The Prime Minister has engulfed the Tory government in sleaze and scandal within days of entering Downing Street. He is distracting the UK government from the essential task of fixing the Tory economic crisis by keeping Ms Braverman in her role.
"The UK government must now ensure maximum transparency and publish all relevant information relating to the Home Secretary's leaks and breaches, including the full advice given to Mr Sunak on her appointment.
"And the Home Secretary must come to parliament to answer MPs' questions. Not just on her leaks - but also on the shocking reports she ignored advice to move people out of Manston, leading to overcrowding, a diptheria outbreak and unlawful detentions."
Yvette Cooper, the shadow home secretary, warned the Government it could not "play fast and loose" with national security, as she repeated Labour’s calls for Rishi Sunak to sack Ms Braverman.
She said: “You can’t have a Home Secretary who is not trusted by the security service, who is not trusted with important government information.
“We think that the papers and the warnings that were provided by the Cabinet Office and by the Cabinet Secretary to the Prime Minister should be sent to the Intelligence and Security Committee.
“So far, we’ve been asking repeatedly whether the Home Secretary has used her personal phone to send other government documents.
“There’s also questions about whether she was investigated for other security leaks, including around a case involving the security service, and around a case involving sensitive legal advice around Northern Ireland.”
Ms Braverman – dubbed “Leaky Sue” – was reportedly previously investigated by Government officials after the leaking of a story involving the security services.
The Daily Mail reported that MI5 played a role in the inquiry after the leak at the time Ms Braverman was attorney general sparked “concern” in the security service.
It come with Home Secretary also under fire over allegations that she ignored advice that migrants were being detained for unlawfully long periods at the Manston asylum processing centre in Kent, claims the Home Office dismissed as “completely baseless”.
The Liberal Democrats have called on the Government to publish the advice reportedly ignored by Ms Braverman.
Downing Street insists Ms Braverman has the "full confidence" of Mr Sunak.
It came on a day that saw Mr Gove fail to deny an incendiary report that Ms Truss’s personal phone was hacked by Russian spies/
Asked about the allegations, Mr Gove told Sky News’s Sophy Ridge on Sunday programme: “I don’t know the full details of what security breach, if any, took place.
“What I do know is that the Government has very robust protocols in place in order to make sure that individuals are protected, but also that Government security and national security are protected as well.”
He said he could not discuss national security matters, as “loose lips can sink ships when it comes to these questions”.
The former head of the British Army said Ms Truss’s use of a personal phone pointed to “ill discipline”.
Lord Dannatt told Times Radio: “Our leaders must be sufficiently disciplined to only communicate through authorised means which themselves are encrypted and are secure.
“We’ve seen it with Suella Braverman, apparently sending messages that she shouldn’t have done on a personal email, and now we get it with Liz Truss.
“This, frankly, is not good enough.”