Douglas Gordon has directed Neck of the Woods, starring Charlotte Rampling, at the Home theatre in Manchester.
Critics have described it as a "vanity project" and "humourless and sedate".
The show features several axes, and Gordon is thought to have wielded an unused prop to take a chunk out of the wall, which he then signed and dated.
He inflicted the damage on Saturday, the day after Neck of the Woods opened as part of the Manchester International Festival (MIF).
The show begins with the sound of an axe, and the stage has a number of axes screwed to it.
The Daily Telegraph said Neck of the Woods had "the unmistakable whiff of a vanity project", with a script that "simply isn't very good", while "Rampling looks terribly uncomfortable most of the time".
The Guardian, meanwhile, described it as a "humourless and sedate Red Riding Hood retelling" that "takes itself very seriously" and is "so old-fashioned you wonder if Gordon has any familiarity at all with contemporary theatre".
MIF artistic director Alex Poots said: "We understand that one of our artists acted in a wholly inappropriate way on Saturday night, causing slight damage to the fabric of Home's new building.
"This is totally unacceptable, and the artist involved will be paying for repairs.
"MIF and the artist have contacted staff who were present and our co-commissioning partners at Home to apologise."
Mr Poots went on to say that the festival supported artists to make ambitious shows.
He added: "We do not support or condone reckless, inappropriate or intimidating behaviour and will work with our co-commissioning partners and artistic and producing teams to ensure that this doesn't happen again."
Gordon won the Turner Prize in 1996.
Home opened in May after being built at a cost of Â£25m.
A Turner Prize-winning artist has used an axe to attack the wall of a theatre where he has staged a new play to scathing reviews.