The Gupta family with close ties to President Jacob Zuma and the ruling ANC party landed a chartered plane at Waterkloof Air Force Base near Pretoria on Tuesday without permits, prompting a national condemnation.
"In light of what happened... the family would like to issue a general apology to all affected, including the South African and Indian governments, the local authorities, the South African public and especially our guests," said Atul Gupta in a statement.
A senior South African diplomat and several police officers have been suspended for their role in allowing the aircraft to land at the restricted base.
Some 200 passengers, including Bollywood stars and politicians, were attending the lavish wedding of Vega Gupta, a niece of the powerful Gupta businessmen who own several companies including Sahara Computers and the pro-government New Age newspaper.
They received a full police escort to the Sun City resort for the four-day affair billed as "the wedding of the year".
In a bid to save face, South African authorities on Friday announced a full scale seven-day probe into the controversial landing, which has been described as an embarrassment and undermining the country's laws.
India's High Commissioner to South Africa, Virenda Gupta, has also been implicated in the scandal, as he is believed to have asked for landing rights, although it's not clear who granted the permission.
The foreign ministry's chief of state protocol Bruce Koloane was Thursday suspended pending an investigation into the matter.
Earlier in the week, the Gupta family said it had received permission for the aircraft -- which was carrying "foreign ministers and other dignitaries" -- to land at the base.
The wedding party left the country on Friday evening through OR Tambo International Airport, after the jet was ordered to vacate the base.