Udo Landbauer, lead candidate for the Freedom Party (FPÖ) in elections in Lower Austrian state last Sunday, said he was giving up all political functions including as a local deputy.
Landbauer, 31, who was also head of the FPÖ's youth wing the RFJ, said he had been trying for days to refute all accusations against him.
But with his local party branch "under siege", Landbauer he said he was giving in to a "media witch-hunt" in order to "take my family out of the firing line".
He also suspended his membership of the FPÖ, which since December has been part of Austria's ruling coalition, and said he was going on holiday, the Austria Press Agency reported.
Last week the Falter weekly reported theat Landbauer's student fraternity had produced a song book in 1997 that included lyrics such as "Step on the gas… we can make it to seven million".
Six million Jews were killed by the Nazis in the Holocaust during World War Two, many of them in gas chambers at industrial-scale extermination camps such as Auschwitz-Birkenau.
Several leading members of the FPÖ – a party created by former Nazis in the 1950s – belong to student fraternities, some of which believe in a "Greater Germany" to include Austria.
Chancellor Sebastian Kurz, under pressure over the furore, said on Wednesday that his government would initiate proceedings to dissolve the fraternity Germania zu Wiener Neustadt.
The FPÖ says the fraternities are harmless, with its leader Heinz-Christian Strache saying on Friday that "anti-Semitism, totalitarianism (and) racism are the opposite of fraternity thinking".
Landbauer, who was 11 years old when the book was printed, has said that he was unaware of the offending text until last week. He suspended his membership after the Falter report.
The affair also embarrassed the centre-left Social Democrats (SPÖ) after it emerged that a party member – one of four people under investigation by prosecutors – had illustrated the song book.
Harald Vilimsky, FPÖ general secretary, said that the "irreproachable and upstanding" Landbauer was the "innocent" victim of a "political and media frenzy".