The squall blew up after it was reported that the shadow home secretary told Mr Davis to "f*** off" after he seemed to lean in to try to embrace her in a Commons bar after MPs voted overwhelmingly to trigger Article 50 and formally begin withdrawal talks with the EU.
After the incident, the Brexit Secretary exchanged texts with a Tory colleague regarding Ms Abbott, according to the Mail on Sunday.
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The newspaper reported that the friend texted Mr Davis, saying: "Cannot believe you made an attempt to give DA (Diane Abbott) a hug!"
The Cabinet minister replied: "Didn't, but the myth grows. I whispered in her ear 'Thanks for your vote' hence the 'F off'. I am not blind."
The colleague then wrote: "Ha! Ha! Thank God you aren't blind. Great week for you and Brexit!"
Then, in an apparent reference to the Specsavers advert in which people with bad eyesight make embarrassing mistakes, Mr Davis texted: "Actually it would make a good Optical Express advert ... Yes, a reasonable success."
A spokesman for the Brexit Secretary insisted Mr Davis respected Ms Abbott and had been joking, saying: "This was a self-evidently jocular and private exchange with a friend.
"The Secretary of State is very sorry for any offence caused to Miss Abbott, someone he has known and respected for many years."
Labour MP Jess Phillips said: "You'd have hoped this sort of misogynistic, sexist attitude had gone out in the 1950s."
Labour MP Chuka Umunna told Sky's Sophy Ridge: "I think it's sexist, I think it's misogynistic... It's appalling.
"This type of behaviour has absolutely no place in the Conservative Party or British politics full stop.
"David Davis, he is a member of the cabinet. This is a member of the cabinet. What does that say about that party to the country, your viewers, if he is coming out with that sort of thing about a very respected long-serving member of the House of Commons?"