South Korea were beaten 2-0 by Jordan at the tournament earlier this month and it later emerged that skipper Son had dislocated a finger in an altercation with Lee on the eve of the semi-final.
Both players released social media statements on Wednesday, with Tottenham's Son posting a picture of the pair together, smiling and with arms around each other's shoulders.
The 23-year-old attacking midfielder Lee, who has been blamed by fans for the fracas and dumped by advertisers, said that he had travelled to London to apologise to Son in person.
Addressing the controversy publicly for the first time, Son wrote in a statement on Instagram: "I am sincerely sorry for causing trouble and will work harder to make the national team grow further."
The incident at the Asian Cup reportedly happened after some younger players in the national squad, including rising star Lee, rushed through their meal so they could leave early to play table tennis.
This displeased Son and some of the older players because meals on the eve of big games are traditionally seen as a time for bonding.
As tension grew Lee tried to punch Son, local media reported. Lee's representatives denied a punch was thrown.
The altercation was cited as a factor in the dismissal last week of coach Jurgen Klinsmann after 12 months in the post.
"I think my actions could be criticised," Son wrote.
But he added: "I would do it again for the team if the same situation arrived because it's one of the captain's duties."
Lee has become public enemy number one since details of the flashpoint emerged, but Son said it was time to move on.
"Kang-in has been going through a very difficult time after the incident," Son wrote. "Please forgive him with an open mind.
"Kang-in is sincerely reflecting on his actions and has apologised to me as well as every national team member."
Last week, Lee apologised and did so again on Wednesday.
"My short-sighted thoughts and improper behaviour have let down Son, the national team and football fans," Lee said in an Instagram post published around the same time as Son's.
"Through a long conversation I came to understand the heavy responsibility that weighs on Son as captain and reflect on myself."
