Warner told a news conference on Thursday that he was sorry his actions had let down his team mates, the fans and Cricket Australia, and apologised to Root.
"I struck Joe Root in the face and I'm here to apologise to Joe and own up to it," Warner said.
"I'm extremely remorseful. I have let my team-mates, Cricket Australia, the fans, myself and my family down.
"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."
Earlier, the batsman was suspended until the first Ashes test on July 10 and fined 11,500 Australian dollars (£7,000) for his part in the incident which occurred in the early hours of Sunday in Birmingham after Australia's Champions Trophy defeat by England.
Media reported that the "unprovoked physical attack", in the words of the England and Wales Cricket Board, was triggered by Root wearing a fancy dress wig in Australia's green-and-gold colours as a beard.