Saatchi is believed to have accepted a caution for assault after going to a police station in the capital on Monday afternoon.
In a statement on Monday night the Metropolitan Police refused to name him, but said: "Officers from the Community Safety Unit at Westminster were aware of the Sunday People article which was published on Sunday 16 June and carried out an investigation. This afternoon, Monday 17 June, a 70-year-old man voluntarily attended a central London police station and accepted a caution for assault."
The former advertising executive said he was holding the broadcaster and cook by the neck "to emphasise my point" as they discussed family, he told London's Evening Standard. But he admitted that the pictures, in which his wife appears to be grimacing as he holds her outside Scott's in Mayfair, looked "horrific".
He told the newspaper: "About a week ago, we were sitting outside a restaurant having an intense debate about the children, and I held Nigella's neck repeatedly while attempting to emphasise my point.
"There was no grip, it was a playful tiff. The pictures are horrific but give a far more drastic and violent impression of what took place. Nigella's tears were because we both hate arguing, not because she had been hurt. We had made up by the time we were home. The paparazzi were congregated outside our house after the story broke yesterday morning, so I told Nigella to take the kids off till the dust settled."
A spokesman for Scotland Yard said no one had come forward to complain to the police.
There was no sign of the couple at their west London home or at Saatchi's gallery, near Chelsea's King's Road, on Monday.
A spokesman for Lawson confirmed she left the family home, in a multimillion-pound converted factory, with her son on Sunday, but refused to say whether it was a temporary or permanent move.