The controversy clouded Horton's Saturday triumph in the 400 meters freestyle final over Sun, the London 2012 defending champion, with the latter suggesting Horton had made the remarks to affect him. Sun won the silver medal.
"You have offended against the sensitive (feelings) of the Chinese people," said one of more than 300,000 comments left on Horton's latest Instagram post of him celebrating his win on the medal podium.
"Apologise to Sun Yang!", said another on Horton's Facebook page that was peppered with insults, derogatory pictures, many of which carried the hashtag "#apologisetosunyang".
Many of the comments recalled a 2013 incident when the Australian men's freestyle relay team were suspended for using a sedative banned by their national Olympic committee (AOC) in a bonding session before the London Games.
The 20-year-old Australian had acknowledged after their final that there were tensions between him and Sun after a splashing incident during a recent practice.
"I used the words 'drug cheat' because he tested positive," he said.
A spokesman for the AOC said Horton was entitled to express his point of view.
"He has spoken out in support of clean athletes. This is something he feels strongly about and good luck to him," the spokesman said in an email to Reuters.
In 2014, it emerged that Sun had secretly served a three-month suspension after he tested positive for a banned stimulant. He said at the time the stimulant was in medication to treat a heart issue and did not enhance his performance.
Sun, the first Chinese swimmer to win an Olympic gold medal, is very popular at home but has been the subject of multiple controversies. In 2013, he spent a week in jail for crashing a car when driving without a license.
He gained widespread sympathy at home on Saturday after breaking down in tears following his loss. Chinese media carried pictures and videos of him crying and hugging a friend with the phrase "Sun Yang don't cry" trending across social media.
Banner image credit: Reuters