Ms Cooper declined to serve in Mr Corbyn’s frontbench team after coming third to him in the leadership election of 2015, but maintained a high profile in the House of Commons as chair of the influential home affairs committee.
A minister under both Tony Blair and Gordon Brown, the Normanton, Pontefract and Castleford MP was seen as a potential candidate for those seeking a return to the agenda of the New Labour years.
But she said she recognised that her criticisms of Mr Corbyn’s leader would prevent her from being the unifying candidate which the party needs.
Writing in The Guardian, Ms Cooper said: “I stood against Jeremy last time and, while I agreed with him on tackling inequality and climate change, I criticised him on security and credibility.
“I was proud to be a minister in the last Labour government – a government that some current shadow cabinet members have strongly criticised.
“As a result, whether fair or not, I recognise there are many in our party who won’t see me as the person to pull all sides of the party together following Jeremy’s departure. And rebuilding towards a Labour government is more important than any one person.
“So I am really grateful to everyone who urged me to stand again but this isn’t the time for me to do so.”
Ms Cooper ruled out quitting as an MP after 22 years, saying she was “not going anywhere”.
“The next leader needs support and we all need to keep fighting for the Labour government our country so badly needs,” she said.
In a clear swipe at Mr Corbyn, who claimed after last month’s election drubbing to have “won the arguments”, Ms Cooper said it was essential for Labour to face up to the scale of its defeat. And she rejected the leadership claim that it was Brexit which cost Labour the election.
“We didn’t win the argument, we lost,” she said.
“Our socialist party lost the majority of the working-class vote and fell behind in the very industrial towns that forged Labour and were hardest hit by Tory austerity. Labour values on equality, solidarity and social justice matter more than ever, and many people are crying out for a progressive government. But we couldn’t persuade people this time and unless we get our act together, we could lose even more support in future.”
Ms Cooper warned that “a gulf has grown between the party and the voters we lost” and said the next leader “has to bridge that gap, but they won’t stand a chance of doing so if the rest of us pretend it doesn’t exist”.
She warned that Labour would not win as a “narrow hard-left party” or as a party consumed by “polarised factional infighting”.
Labour must be “a party for the whole country, not just a liberal-labour party for the cities” and must learn to be proud of the achievements of the Blair and Brown administrations, she said.
And she said it cannot focus only on opposing Boris Johnson, but must also show it is “credible as an alternative government”.
A key priority for the new leader must be to take “strong action against antisemitism” and bring “kindness and integrity” back into politics, said Ms Cooper.
In a message to those expressing doubt that Labour can ever again win power, she said: “We are further behind than in 1992 and face bigger challenges, not least in Scotland. But we also still have great strengths and passion, amazing activists who worked hard for every vote and believe in a better Britain.
“And it feels like things change faster now so who knows what we can achieve if we try. So join. Or rejoin. Be part of this.”