Cardona was caught on camera appearing to make a slant-eyes gesture during a minor scuffle between the two sides in the second half of Friday's match at Suwon World Cup Stadium, south of Seoul.
"As regards to what happened today, it was not my intention to disrespect anybody, or a country or race, but if anybody interpreted it that way, I am sorry," Cardona said in a video posted on the Colombian team's Twitter account.
"I am not an aggressive person and that moment in the match has been badly interpreted," the 24-year-old said.
South Korea captain, midfielder Ki Sung-Yueng, condemned Cardona's actions, saying, "racist behavior is absolutely unacceptable".
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"Colombia have world-class players, and it was disappointing for the team to show such actions," Ki added.
The Colombian football association "sincerely" apologised over the incident to its South Korean counterpart, Yonhap news agency reported.
In response, Korea Football Association (KFA) expressed a "deep regret" over the episode and called for a "proper punitive measure" against Cardona, South Korea's Yonhap said.
The controversy is not the first time South American players have been embroiled in race-related rows in Asia.
Shanghai SIPG's Brazilian star Hulk was investigated earlier this year over an alleged racially motivated assault on a rival team's coach, a claim both he and his club deny.
Days after that incident, Argentina's Ezequiel Lavezzi, who plays for Hebei China Fortune, sparked an uproar when promotional photos emerged of him pulling back the corners of his eyelids in a "slant-eyed" pose.
South Korea won Friday's match 2-1.