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Warner, 26, was suspended until the first Ashes Test on 10 July and fined £7,000 (AU $11,500) after the incident early on Sunday morning.
"I struck Joe Root in the face and I'm here to apologise to Joe and own up to it," Warner told a media conference.
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.
"I'm extremely remorseful. I have let my team-mates, Cricket Australia, the fans, myself and my family down."
Warner will now miss the remainder of his country's Champions Trophy campaign and the tour matches against Somerset and Worcestershire, although he is eligible for the first Test at Trent Bridge.
He added: "I have accepted the punishment that has been handed down and I'm accountable for my actions.
"Since then I have exchanged text messages with Joe. He accepted my apology which I am thoroughly grateful for."
Warner, who has played in 19 Test matches for his country, was dropped for Australia's Champions Trophy game against New Zealand on Wednesday as a punishment.
He could have been sent home from the tour after pleading guilty to a charge of "unbecoming behaviour".
Australia captain Michael Clarke admitted Warner's behaviour was unacceptable.
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"Although the punishment for David is quite harsh that's the reality when you play for the Australian cricket team," said Clarke.
"This is not an IPL team, this is not state cricket, it's not county cricket. When you play for Australia there are standards you have to uphold. His actions have not met the standard required of an Australian cricketer."
Warner's suspension means he will miss valuable match-practice as he aims to end a poor run of form.
He has made seven single-figure scores in his last nine innings while playing for Delhi Daredevils in the Indian Premier League and one-day internationals for Australia.
This incident is not the first time Warner has been in trouble with Cricket Australia, and last month he was fined £3,700 for a foul-mouthed Twitter exchange with two Australian journalists.
Warner admitted he needed to improve his behaviour and said: "I've got to be a bit smarter with what I do on and off the field. Hopefully I will still have a part to play in the Ashes."

The Virgin East Coast driver sheltered next to his cab just three seconds before the high-speed train passed.
The 3 August incident happened 12 miles (19.7km) north of York station.
A report said it was the result of a lack of "clear understanding" between the driver and a trainee signaller over whether a line blockage was in place.
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The Rail Accident Investigation Branch report said the driver was attempting to carry out tests on the 07.55 BST Inverness to London King's Cross train when it experienced a loss of power at Kyle Beck, near Tollerton.
It said the driver contacted the signaller - a trainee working under the guidance of an experienced colleague - to request trains on the adjacent lines be stopped in order to carry out the checks.
After communicating the request, the signaller said he would call the driver back to confirm the line blockage.
But, the report found, the driver stepped down from his cab before receiving the confirmation.
A CCTV image taken from the front of the passing train showed the driver sheltering next to his vehicle. The driver was unhurt.
"It is possible the driver may not have heard what the trainee signaller said or that he had misunderstood what he was being told," the report said.
"The driver and the trainee signaller did not reach a clear understanding about stopping trains on the adjacent lines and the driver subsequently alighted from the driving cab of his train, incorrectly believing it was safe to do so."
Australia batsman David Warner has apologised for punching England's Joe Root in the face in a Birmingham bar.
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A train driver narrowly avoided being hit by a second train travelling at 105mph (169km/h) when he was investigating a power failure.