Warner, 26, was also fined £7,000 (AU $11,500) after the incident, which took place in Birmingham on Sunday morning.
Jim MaxwellAustralian Broadcasting Corporation commentator
"A kid from the back blocks with a strong personality, Warner is outgoing, likes talking to people and always has an opinion. He is popular with his team-mates and respected in the dressing room. He has that lively and nocturnal side to his personality. But he is 26 now, and off the field he needs to change."
He will now miss the remainder of his country's Champions Trophy campaign.
Warner will also be absent for tour matches against Somerset and Worcestershire, but can play in the first Ashes Test on 10 July.
England and Australia players were in the Walkabout pub in the early hours of Sunday when opening batsman Warner is alleged to have struck England batsman Root.
BBC Radio 5 live's Pat Murphy said: "I believe it happened around 02:00 (BST) at the bar in the centre of Birmingham.
"There was a private VIP area and some of the Australia players were in there. The three England players, Joe Root, Stuart Broad and Chris Woakes, were some way away and were larking around - wearing silly wigs and that.
"I understand Warner took umbrage and went over to Root and glanced him on the chin.
"Broad put his arm around Root as soon as it happened and said 'time to go back to the hotel' and I am also told that they were not worse for wear."
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