Tory MP Nadhim Zahawi says Labour leader Ed Miliband should sack her for "inciting hatred against white people".
Ms Abbott, the first black woman to be elected as an MP, later wrote her remark had been "taken out of context".
She was reacting on Twitter to a conversation about media coverage in the wake of the Stephen Lawrence murder trial to a woman who complained about the use of the term "the black community" in the media.
The Conservative MP Nadim Zahawi said Abbott should apologise or resign after reading he read the message, which was posted as part of an exchange with a freelance journalist about the merits of talking in general terms about the black community.
Abbott said on Thursday that her remark had been misunderstood. In another post on Twitter, she said: "Tweet taken out of context. Refers to nature of 19th century European colonialism. Bit much to get into 140 characters. "
She used the same phrase later when asked about the row in an interview with Sky, saying: "I think the tweet was taken out of context and some people have interpreted it maliciously. "
Mr Neil immediately reacted, asking whether she thought “black mums love their kids more than white mums”.
Furious, the MP retorted: "'I have said everything I am going to say about where I send my son to school. "
Born in London to Jamaican immigrants, Miss Abbott read history at Cambridge University before becoming an administration trainee at the Home Office.
She worked as a race relations officer for the national Council for Civil Liberties before moving into the media, joining Thames Television and later TV-am.
She was a press officer at the Greater London Council and Lambeth Council before changing tack and moving into politics.
After a brief stint at Westminster City Council, Miss Abbott was elected into the House of Commons in 1987, becoming the UK’s first black woman MP.
She quickly garnered a reputation as a maverick who was unafraid to speak her mind and could not always be relied upon to toe the party line.
She was outspoken in her condemnation of Tony Blair’s policies on Iraq and tuition fees and was a stern critic of Harriet Harman when she opted to send her children to a grammar school in 1997.
Adewunmi said she objected to the way in which the term the "black community" was used as a generalisation.
Abbott responded with a series of messages. "I understand the cultural point you are making. But you are playing into a "divide and rule" agenda, " she said. Then she added: "White people love playing "divide & rule" We should not play their game #tacticasoldascolonialism. "
And finally she said: "Ethnic communities that show more public solidarity & unity than black people do much better #dontwashdirtylineninpublic. "
Although her comments were addressed to Adewunmi, Abbott did not use the Twitter direct message facility which keeps messages private and, as a result, they were available for anyone to read.
On Thursday, as the remark circulated widely on Twitter, Zahawi, the MP for Stratford-on-Avon, accused Abbott of racism.
"A healthy society should not tolerate any form of racism. DAbbott should apologise and resign or EdM must sack her, " he wrote on Twitter.
The Labour party was not commenting on the row on Thursday, but Abbott did receive backing from the former Labour MP George Galloway. "Diane Abbott has been my friend for 25 years; only the obtuse would think her a 'racist', " Galloway posted on Twitter.
The MP was investigated by the Committee on Standards and Privileges in 2004 when it emerged she had failed to declare earnings of £17, 300 for appearances on The Week. The complaint was upheld and she was forced to apologise to the House.
She was also criticised during the MPs expenses scandal, after claiming £142, 000 in 2009, including £1, 100 on taxis.
When she joined the 2010 Labour leadership race, many were surprised. But she insisted that at least one candidate should be female and an ethnic minority, saying: “If not now, when? And if not me, who? ”
She eventually lost out to Ed Miliband but the contest only served to enhance her public profile.
Miss Abbott was a regular guest on The Week, appearing alongside former Tory MP Michael Portillo, whom she had known since school and with whom she once appeared in a joint school production.
She has also appeared on Have I Got News For You, Celebrity Come Dine With Me and Celebrity Cash in the Attic.
MP Diane Abbott in Twitter race row
Thursday, January 5, 2012, 8:36 AM
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