The former Downing Street spin doctor said in a speech on Saturday that “huge swathes of the country” cannot accept “this Labour party” in power.
His intervention comes after Jeremy Corbyn’s party made some gains in the local elections but failed to make a breakthrough in key swing seats.
Mr Campbell told a conference hosted by the centre-left pressure group Progress: “If we cannot beat this shambles of a Tory party, we don’t deserve to be in the game.
“My Labour tribalism is being pushed to the limit – by the return of militant-style nastiness in local politics.
“By my revulsion that any anti-Semitism has been allowed to fester.
“By the feeling that some in the leadership, and their supporters, feel much greater animus against other Labour supporters than against Tories.”
Mr Campbell, who served as communications director under Tony Blair, also referenced a soundbite of the era to say Labour should be “tough on Brexit, tough on the causes of Brexit”.
He added that if Labour’s Brexit tests are to mean anything, the party cannot support Prime Minister Theresa May’s strategy.
“If they do, then though the Tories will be seen as Brexit’s architect, Labour will be its bricklayer and history harsh when the house comes tumbling down,” he said.
Labour MP Stephen Doughty told Sky News that “both the parties would have liked to have done better” in Thursday’s poll.
He welcomed Labour’s clarity on remaining in a customs union after Brexit, adding: “I want to see us go further and back a people’s vote.”
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Mr Corbyn has hailed the local election results as “building on the historic gains” Labour made in last year’s general election.
He predicted that with activists’ support “we can and will form a government”.