The apology came a day after Hong Myung-bo resigned as South Korea coach. Results on Saturday meant the South Koreans would not advance to the last 32 as one of the eight third-placed finishers in the group phase.
Expectations had been high for a talented South Korean squad featuring Son, Paris St Germain midfielder Lee Kang-in and Bayern Munich defender Kim Min-jae. Drawn alongside co-hosts Mexico, the Czech Republic and South Africa, they were widely expected to reach the knockout rounds.
South Korea opened with a 2-1 win over the Czech Republic, but defeats by Mexico and South Africa left them reliant on results elsewhere, ultimately falling short of a place in the knockout stage.
"I don't know where to begin," Son said in a lengthy Instagram post. "I can't pretend I don't know what happened, nor do I want to escape reality.
"First and foremost, I would like to offer my sincere apologies to the people of South Korea and to all the fans who love soccer.
"As a soccer lover myself, I can only imagine how heartbreaking, frustrating, and painful it must have been for you to watch a match like this. That is why I feel that a simple 'I'm sorry' cannot begin to convey the disappointment and hurt you must be feeling - it feels woefully inadequate even to say it..."
The former Tottenham Hotspur player, now 33, endured a difficult tournament, failing to score or provide an assist in any of South Korea's three matches.
Questions have also been raised about Hong's handling of the winger. Son started on the bench for South Korea's decisive clash with South Africa and was introduced only in the second half. Hong had also withdrawn him during the defeat by Mexico.
Son, who had previously considered ending his international career after South Korea's semi-final defeat at the 2024 Asian Cup, earlier this month dismissed speculation that the tournament in North America would be his last World Cup.
"I will give it my all to bring joy to you all once again. I have never forgotten the promises I made to our fans. Until you call for me, until you need me, I will pour my heart and soul into preparing myself to do better," Son said.