The Rotherham MP wrote in the Sun on Friday that "Britain has a problem with British Pakistani men raping and exploiting white girls".
She has now apologised for her "extremely poor choice of words" and quit as shadow equalities minister.
Jeremy Corbyn said Labour would not "demonise any particular group".
Ms Champion's article was written after 17 men were convicted of forcing girls in Newcastle to have sex.
The men, who were mostly British-born, were from Iraqi, Bangladeshi, Pakistani, Indian, Iranian and Turkish communities.
In interviews following the convictions, the Labour MP said such crimes involved "predominately Pakistani men" and said a fear of being called racist was hampering the authorities' investigations.
She then wrote the Sun article, which also included the line: "These people are predators and the common denominator is their ethnic heritage."
Among the article's critics was fellow Labour MP Naz Shah, who said it was "irresponsible" and "setting a dangerous precedent".
Following the backlash, Ms Champion sought to distance herself from the article, claiming it had been altered, something denied by the newspaper.
Announcing she was stepping down, Ms Champion said: "I apologise for the offence caused by the extremely poor choice of words in the Sun article on Friday.
"I am concerned that my continued position in the shadow cabinet would distract from the crucial issues around child protection which I have campaigned on my entire political career.
"It is therefore with regret that I tender my resignation as shadow secretary of state for women and equalities."
Mr Corbyn said he accepted Ms Champion's resignation, saying his party would not "blame any particular group or demonise any particular group [and] the issue is one of safety of individuals."
He added he was looking forward to working with her in the future because she is "totally committed to protecting vulnerable women".
The MP's Sun column followed an interview on BBC Radio 4's Today programme in which she warned that people were not raising potential cases of child abuse as they feared being labelled racist.
She wrote: "Britain has a problem with British-Pakistani men raping and exploiting white girls.
"There. I said it. Does that make me a racist? Or am I just prepared to call out this horrifying problem for what it is?"
In the newspaper, the column was accompanied by another article by former Sun political editor Trevor Kavanagh, which referred to "the Muslim problem".
Mr Kavanagh's article was condemned in an open letter signed by more than 100 MPs, including Ms Champion herself, who said she was horrified that an "extreme Islamaphobic" column had quoted her.
A spokeswoman for the Sun said: "Sarah Champion's column, as it appeared on Friday, was approved by her team and her adviser twice contacted us thereafter to say she was 'thrilled' with the piece and it 'looked great'.
"Indeed, her only objection after the article appeared was her belief that her picture byline looked unflattering. Her office submitted five new pictures for further use."
Former UKIP leader Nigel Farage said in a tweet: "Telling the truth about tough issues is now impossible in the modern Labour Party. Pathetic."
Rebecca Hilsenrath, chief executive of the Equality and Human Rights Commission said it was a "real shame that a respected advocate of equality" had stepped down over, what it described as an "oversensitivity about language".
Labour shadow minister Sarah Champion has quit the party's front bench after criticism over a newspaper article she wrote about grooming gangs.