The party took out a full-page advertisement in a national newspaper to insist Ukip was not racist but repeated its warning about the risk posed by organised criminal gangs from Romania.
Mr Farage said his comments about people being right to be concerned if a group of Romanian men moved in next door had caused a “predictable storm of protest and accusations of racism”.
The Ukip leader initially stood by his remarks, which came during a bruising interview with LBC, but last night he said: “Do you know what, in life sometimes people get things wrong.”
He told BBC News: “I regret the fact that I was completely tired out and I didn’t use the form of words in response that I would have liked to have used.
“I should have just hit back immediately and said: ‘Look, understand there is a real problem here - you can’t deny it - too much criminality from these gangs has come to London’.”
In an advertisement in The Daily Telegraph taking the form of an open letter from Mr Farage, the Ukip leader said: “Let me be clear - Ukip is not a racist party, and our immigration policy, far from being racist, aims to end discrimination against non-Europeans.
“The vast majority of Romanians who have come to the UK wish to better their lives and would make good neighbours.
“But there is a real problem, an unpalatable truth that our political class would rather not discuss. Since the welcome fall of Communism and the awful dictator Ceausescu, Romania has struggled to complete a full transition into a western democracy.”
There was discrimination against the Roma minority and a “huge problem” with the growth of criminal gangs, he said.
Mr Farage claimed European Union free movement rules meant there was “nothing the UK authorities can do to stop such people from entering our country”.
He said: “We should not be in a political union with Romania, with an opened door to all of their citizens.”
By leaving the EU and “taking back control of our borders” the “necessary checks” could be done on would-be immigrants.
“When this happens my answer to the question ‘should people be concerned if a group of Romanian men moved in next door?’ will be ‘no’.”
Deputy Prime Minister Nick Clegg said that Mr Farage’s remarks had revealed his “divisive, nasty approach” to politics.
“I think the mask is starting to slip and I think what’s being revealed that sort of behind the beer-swilling bonhomie is a rather nasty view of the world,” he told BBC1’s The Andrew Marr Show.
“I think anyone who singles out one community, one nationality, and says ‘I don’t want to live next door to them’, I really think that’s the politics of division and I think it really should have no place in modern Britain.”
Labour leader Ed Miliband described Mr Farage’s original comment on Friday as a racial slur that was “completely out of order”.
Mr Miliband said he could understand why people supported Ukip because they are not happy with the political system, but he stressed Mr Farage’s party do not have the solutions.
He told BBC Radio 4’s Today programme: “I think though our politics is sort of disagreeable enough without political leaders saying about other political leaders ‘they are a racist’.
“I think it was deeply offensive, I think it was wrong what he said.
“I don’t know Nigel Farage very well, I think he made one remark which was completely wrong, completely out of order, and was a slur.
“Apparently he’s said he got it wrong, he should certainly say he’s got it wrong.”
Asked whether he understood why people would vote Ukip, Mr Miliband said: “I can understand why people feel deep discontent with the political system and why people would look at other alternatives, of course I can.
“My view is that Ukip might be siding with the discontent that people feel but they don’t have the solutions.”