That investigation has now taken place and Mr Hatzidakis will face “no further action”, the Football Association (FA) said in a statement.
“We have thoroughly reviewed all of the evidence in relation to the recent incident at Anfield involving the Liverpool defender Andrew Robertson and match official Constantine Hatzidakis, and we will be taking no further action,” a spokesperson said.
“Our comprehensive process involved reviewing detailed statements from Liverpool and PGMOL, as well as multiple angles of video footage, in relation to both the incident and its surrounding circumstances.”
In a statement released by the PGMOL, Mr Hatzidakis said: “I fully assisted The FA with their investigation and have discussed the matter directly with Andy Robertson during an open and positive conversation.
“It was certainly not my intention to make any contact with Andy as I pulled my arm away from him and for that I have apologised. I look forward to returning to officiating matches.”
The official could be seen reacting after being confronted by Robertson shortly after the half-time whistle.
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Sky Sports cameras appeared to show him throwing up his arm and making contact with Robertson, who gesticulated that he had been hit by the official.
Liverpool boss Jurgen Klopp said he did not see the incident, but the Sky Sports pundits working on the game reacted with incredulity.
Gary Neville, the former Manchester United and England full-back, said: “I’ve never seen an official raise an elbow to a player. I think he’ll be in a lot of trouble after this game ends.”
Ex-Manchester City and England defender Micah Richards added: “I’ve never seen anything like that in my life.”
Former Manchester United captain Roy Keane focused on Robertson’s role, calling the Scot “a big baby”.
“He should be more worried about his defending,” said Keane. “Do you know what he is, that Robertson? I’ve watched him a number of times, he is a big baby, that’s what that guy is.
“Just get on with the game and concentrate on your defending. He does grab the linesman first.”