Brook was speaking for the first time since news broke that he had been disciplined with a fine for the incident in Wellington.
The incident occurred in the final week before England departed to Australia for the Ashes tour, which they eventually lost 4-1.
Brook said he reported the incident to England's management and came close to being sacked as captain.
Speaking ahead of the first ODI against Sri Lanka in Colombo, Brook said: "Obviously I made a terrible mistake.
"Not only as a player, but as a captain. It's very unprofessional and I should be leading from the front.
"I've learnt from my mistakes, I've reflected a lot on what's happened and I know it wasn't the right thing to do.
"I want to say sorry to my team-mates, to all the fans that travel far and wide to watch, spend a lot of money on coming out to watch us play cricket and supporting us, and to the ECB for putting them in a tricky situation, and it'll never happen again.
"I'm extremely sorry. I think I've got a little bit of work to do to try and regain the trust of the players.
"I said sorry to them yesterday. I felt like I needed to say sorry for my actions. It's not acceptable as a player, but as a captain it's really not acceptable to do what I did in New Zealand.
"I'll be the first person to say that. I hold my hands up."
England and Sri Lanka will contest three ODIs, starting on Thursday in Colombo, before playing three T20Is ahead of next month's World Cup.
Former England fast bowler Stuart Broad has come to the defence of head coach Brendon McCullum and captain Ben Stokes, saying the current Test team does not have a drinking culture.
Australia defied pre-series claims that their side was too old by overpowering England 4-1 in the Ashes, with a veteran-heavy team turning "Dad's Army" jibes into a commanding series victory.
Brendon McCullum says he is open to evolve as England head coach following a disastrous Ashes tour in Australia but will not change his methods.
England white-ball captain Harry Brook has been fined by the England and Wales Cricket Board following an altercation with a bouncer in New Zealand in October.
