Austrian Vice Chancellor Heinz-Christian Strache was shown in the video meeting a woman posing as a Russian oligarch's niece in 2017, and apparently offering to funnel state contracts towards a company in exchange for political and financial support.
A new coalition with the Freedom Party may not be possible after the video scandal, be it for lack of votes or backing in Kurz's People's Party.
Although he insisted he did not break any law, Mr Strache admitted to the "extremely embarrassing, drunken affair", which saw him appearing to offer infrastructure contracts to a woman posing as a wealthy potential donor from Russian Federation while in Ibiza in July 2017.
In the video, the source of which the newspapers declined to reveal, Strache and party colleague Johann Gudenus are heard telling the unnamed woman she could expect lucrative construction contracts if she buys an Austrian newspaper and supports the Freedom Party.
Kurz accepted his resignation, Strache said.
Austria could set an election date as soon as the summer, according to national law, "but that could be hard due to school holidays", said Robert Stein, who heads the election desk at the interior ministry.
In the recordings, Strache is seen promising public contracts in return for campaign help and suggesting he could help bring about staff changes in Austria's largest-circulation tabloid, the Krone Zeitung, as well as the part-privatisation of public broadcaster ORF.
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The dramatic developments followed the publication by two German newspapers on Friday of footage from a sophisticated hidden-camera sting months before Austria's last parliamentary elections in 2017. He said he had had no further contact with the woman and she did not donate to his party.
Strache's resignation was a black eye for the populist and nationalist forces who favor tighter European immigration policies.
Mr Kurz announced the snap ballot would take place as a matter of urgency, putting the nation's political future - and his own - in jeopardy.
"You can not draw a Russian link to this clearly ugly incident based on the existing recording", Mr Oleg Morozov, a member of the foreign affairs committee of Russia's Upper House and a ruling party lawmaker, told RIA Novosti state news agency. He conceded his behaviour in the video was "stupid, irresponsible and a mistake". Spiegel didn't disclose how it obtained the video and said it doesn't know the motives of the people who made it.
The video is also a headache for Kurz, whose party is still leading in opinion polls but far short of a majority.
Kurz had repeatedly distanced himself from his far right coalition partners over lesser scandals in the past, mostly involving party officials and anti-Semitism or racism - such as one in which the deputy mayor of Hitler's home town wrote a poem likening immigrants to rats.
At the same time, the chancellor has been reluctant to build bridges to the opposition Social Democrats, with whom his party governed in most previous coalitions since World War II.