The 26-year-old, who was playing in the Indian Premier League when he tweeted the insults, faced a disciplinary hearing on his arrival back in Sydney on Wednesday, charged with "unbecoming behaviour".
Warner was angered that a newspaper had used a picture of him next to an article on corruption in the IPL written by Robert Craddock. Another journalist, Malcolm Conn, was attacked when he entered the conversation in support of his colleague.
"In hindsight, clearly I let my frustrations get the better of me and posted some inappropriate tweets last weekend," Warner said in a CA statement.
"While I disagreed with the story and my image being used alongside the story, I could have chosen my words better and I apologise for any offence that my language may have caused.
"I'll continue to have honest conversations with all my followers and I will be mindful of the language I use in future."
The flamboyant left-hander looks likely to open the batting for Australia in back-to-back Ashes series against England starting in July.
Earlier on Wednesday, he received backing from Australia captain Michael Clarke, who suggested the incident may not harm his chances of one day captaining his country.
"He's apologised and made it very clear it was unacceptable," Clarke said at Sydney airport.
"I think Davey has great potential to be a leader of the Australian cricket team. He's a wonderful guy and wonderful player.
"He's learned from this ... and if he continues to grow as he has over the last few years, there's no reason why he hasn't got the potential to captain Australia one day."
Clarke, who once apologised for not "walking" via Twitter, also said there were no plans to restrict the use of social media by players, although they needed to recognise there was "a line that cannot be crossed".