The incident occurred as the players were leaving the pitch at half-time during the Premier League meeting at Anfield, with Robertson approaching Hatzidakis before he then complained of being caught by the official.
No action was taken at the time but PGMOL, the match officials’ body, said Hatzidakis would not officiate until an investigation had been concluded.
That investigation has now taken place, with the FA saying: “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.
“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 through PGMOL, Hatzidakis revealed he had apologised to Robertson, who was given a yellow card by referee Paul Tierney for his complaints after the incident.
“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.”
After the game, Ref Support UK said Hatzidakis should be held to account if he was found to have committed wrongdoing, but said the rules need changing to stop players from crowding officials.
Martin Cassidy, the chief executive, said: “There are no clear boundaries, therefore we believe there should be an exclusion zone of two metres around match officials to prevent aggressive contact.
“If this zone is breached the referee can issue an in-game sanction.
“That said, match officials are the guardian of standards and must also be held accountable if found guilty of similar behaviour towards a player or any participant of the game.
“Any sanction issued to a match official must reflect the same sanctions issued to player and participant.”