The brand argued that the ad showed "empowering, confident women" in bikinis.
In a letter to the Financial Times, Dacre claimed the number of advertisers with the paper had fallen since Greig's appointment.
The advert was seen on the sides of buses in February this year.
Parents mess up all the time. All parents. But with a low bar comes low expectations.
Under new rules, TV adverts must not include gender stereotypes likely to cause harm or widespread offence.
The website has been accused of being “hyper-partisan” and pushing conspiracy theories.
"If I had young children I wouldn’t want them passing that on the way to school."
Advertising Standards Authority said it was looking into claims the adverts misled readers.
"Saw this on TV last night and genuinely got a little happy tear!"