The contest has been brought about by Tuesday's resignation from Parliament of former Tory MP Patrick Mercer over a cash-for-questions scandal.
Mr Farage said he did not want to look like an "opportunist" by entering the contest, as he did not "have any links with the East Midlands".
He added that he wanted to focus on UKIP's European elections campaign.
Although the Conservatives have a majority of 16,000 in Newark, UKIP is currently doing well in opinion polls.
But John Curtice, professor of politics at the University of Strathclyde, said UKIP would be "trying to succeed on the back of no particular local support at all".
Mr Farage told BBC One's Breakfast: "I haven't had long to think about it but I have thought about it, and we're just over three weeks away from a European election at which I think UKIP could cause an earthquake in British politics, from which we can go on and win not just one parliamentary seat but quite a lot of parliamentary seats.
"For that reason I don't want to do anything that deflects from the European election campaign, so I'm not going to stand in this by-election.
"I want to focus the next three weeks on winning the European elections and also I don't have any links with the East Midlands. I would just look like an opportunist, and I don't think that would work."
Asked whether he had decided not to run for fear of losing, Mr Farage replied: "I have shown some courage over the yearsâ€¦
"It's about choosing the right battles. It's about prioritising and I know that if I were to have said yes to standing in Newark the next three weeks would be dominated by am I going to win, am I not going to win, and we wouldn't be talking about open-door immigration, EU membership and that most of our laws being made somewhere else."
Mr Farage referred to the former leader of the Monster Raving Loony Party, once a fixture at such contests, saying: "I'm not Screaming Lord Sutch. I don't stand in every by-election."
Ken Clarke, seen as the most Europhile of the Conservative members of the cabinet, told BBC Radio 4's Today programme that Mr Farage had been right to decide not to run, saying: "I am not really surprised. Whatever else Nigel is, he is not an idiot, and I don't think he'd have the faintest chance of winning in Newark."
He accused UKIP, which advocates leaving the European Union, of "peddling a total nonsense that our economic problems have been caused by immigration".
Mr Clarke, who is a Nottinghamshire MP, said: "I don't think the residents of Newark, some of whom I know because I used to represent some of the villages there, they're not going to vote for a card, larking about, trying to get protest votes."
For the Liberal Democrats, Chief Secretary to the Treasury Danny Alexander said: "Nigel Farage is clearly frightened to put himself forward to try and actually get a seat in the UK Parliament.
"He's very happily heckling from the sidelines doing his thing in the European Parliament ,and I think many people will look at this and say it's not really that impressive."
Mr Mercer, who has represented the Nottinghamshire constituency since 2001, is due to be suspended from the Commons for six months for allegedly asking questions in Parliament in return for money.
He was filmed by undercover reporters from the BBC's Panorama last year apparently agreeing to set up a parliamentary group to push for Fiji to return to the Commonwealth.
The MP had already said he would not contest the general election next year, having served as an independent since May 2013.
In a short statement, the former soldier said he would not contest the findings of a report into his conduct, to be published on Thursday, which will call for him to be barred from Parliament for six months.
He said he was resigning with "a great heaviness of heart" for the sake of his family, adding: "I am an ex-soldier, I believe that when you have got something wrong, you have got to 'fess up and get on with it."
The MP, a prominent critic of David Cameron, who sacked him as a shadow minister in 2007, said he hoped his successor would be a Conservative.
The party has selected Robert Jenrick to contest Newark. Labour - which held the seat between 1997 and 2001 - has chosen Michael Payne as its candidate. The Liberal Democrats have yet to make a selection.
At the 2010 general election, Mr Mercer won 27,590 votes. Labour came second with 11,438 votes, the Lib Dems third with 10,246 and UKIP fourth with 1,954.
Meanwhile, an ITV/ComRes poll suggests that, of people who insist they will definitely vote in the European elections, 38% say they will back UKIP.
The survey puts Labour in second place on 27%, the Conservatives third on 18% and the Lib Dems fourth on 8%.
ComRes interviewed 2,052 British adults between 25 and 27 April.
UKIP leader Nigel Farage has said he will not stand in the forthcoming Newark by-election.