The video appears to show him discussing government contracts with an alleged Russian investor.
Strache blamed his actions on alcohol and acting like a "teenager", saying his behaviour had been "stupid" and "irresponsible", and that he was leaving to avoid further damage to the government.
"After yesterday's video, I must say quite honestly: Enough is enough," he said.
"The serious part of this (video) was the attitude towards abuse of power, towards dealing with taxpayers' money, towards the media in this country," Kurz said, adding that he had been personally insulted in the footage.
He said he had talked about this with Kurz and that they would discuss next steps on Sunday.
A crowd of thousands with placards and banners have been rallying on the square outside Kurz's office, chanting "Snap elections now!"
Kurz has attempted to distance himself from past scandals surrounding the Freedom Party, mostly ones involving party officials and anti-Semitism or racism, but political opponents called for him to respond to the latest revelations.
"This is the tip of the iceberg," Thomas Drozda, from the opposition Social Democrats, told national broadcaster ORF.
"I expect the chancellor, who evidently has known about this video for 48 hours, and that his coalition partner is drowning in a swamp of corruption, to speak and explain his position."
Kurz said that this was not the first time he had had difficulties with the party.
"Even if I didn't express myself publicly at the time, there were many situations that I found difficult to swallow," he said.
The meeting reportedly took place at a villa on the Spanish island of Ibiza, in a private room with both politicians relaxing on sofas, smoking and drinking.
In the footage, the woman says she is the niece of a powerful Russian oligarch. She offers to buy a 50 per cent stake in Austria's Kronen-Zeitung newspaper and switch its editorial position to support the Freedom Party.
The vice-chancellor also speculates that the Russian's takeover of Kronen-Zeitung could boost support for the party to as much as 34 per cent.
As part of the deal, he suggests the Russian woman "set up a company like Strabag", the Austrian construction firm.
"All the government orders that Strabag gets now, (you) would get," he continues.
Strache also names several journalists who would have to be "pushed" from the newspaper, and five other "new people whom we will build up".
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)