"I've made a huge mistake, a huge error in judgement. I'd firstly like to say a huge sorry to my partner Teagan, her family, my family," Smith said at a media conference in South Africa.
"I'm also sorry to my team-mates, NZRU and the New Zealand fans.
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"My behaviour was unacceptable and if you could respect me and my partner in this situation. I'm just trying to get home to deal with this. Thank you."
Earlier, Smith's partner asked for privacy as she dealt with "a very private matter'' at this time.
In a statement released by the NZ Rugby Union yesterday, Smith's girlfriend Teagan Voykovich asked that media not contact her or her family as they dealt with the situation ahead of them, after the halfback and a woman used a disabled toilet at Christchurch Airport.
The statement said Voykovich was suffering from "some distress'' by attempts to contact her about the incident involving Smith and a mysterious woman.
Smith's aunt has defended the All Blacks star, telling people to "leave him alone".
Feilding woman Caroline Smith said her nephew was just a young guy.
She did not know the star player had a partner, but told Fairfax "he's done really well for himself" in his rugby career.
"I suppose we have the All Blacks up on a pedestal, they have them up so high, but he's just a young guy."
The incident occurred after the All Blacks' 41-13 victory in Christchurch on September 17. All Blacks management were only made aware of the incident this week once they arrived in South Africa for their final Rugby Championship test.
"On the Sunday in Christchurch after the South African test at the Christchurch airport Aaron and a female friend were seen by members of the public to enter the disabled toilets," All Blacks coach Steven Hansen said.
"On our arrival to South Africa on Monday we were informed on this. I had a conversation with Aaron who confirmed the story. As a result of this the leadership group met and were unanimous in the fact the behaviour wasn't acceptable and as a consequence a one-match suspension would be put in place."
"Clearly Aaron is bitterly disappointed in his behaviour and feels he has let the team down but, just as importantly, he feels bitterly disappointed in letting his family and his partner down.
"At his request he has asked to go home to deal with the personal matters that have come with this incident and we are happy to support him with that."
It's yet another off-field controversy for New Zealand Rugby following the Chiefs stripper scandal and the Losi Filipo case in the last month.
New Zealand Rugby chief executive Steve Tew said: "This was really bad decision-making by Aaron which reflects poorly on him, and on the team."
Tew said Smith had failed to meet the standards that New Zealand Rugby and the public expect of players.
"We are taking action to improve the culture of our game and are about to embark on an independent review of respect and responsibility in our professional player environment. We have drafted the terms of reference and are currently talking to prospective panel members. We will provide an update on this shortly."