De Rossi was caught by television cameras mouthing "shut up, shitty gypsy" after the players clashed in what was a hot-tempered Serie A clash on Sunday.
Despite calls from organisations representing ethnic Roma people, De Rossi never faced the threat of a sanction because television evidence is not used retroactively in Italian football and the incident was not sanctioned or noted by the match referee.
The Italy international offered an apology on Tuesday - although not directly to Croatian international Mandzukic.
"I'm sorry to anyone who felt offended," De Rossi said when asked to comment by a presenter from the 'Le Iene' television show.
"I was caught by television cameras saying something I shouldn't have. It's not the first time something like this has been said and it happens every so often on the pitch.
"Football can sometimes bring the worst out in players, although that's no justification. We'll try not to say things like this or, as the coach said, we'll cover our mouths."
Roma coach Luciano Spalletti did not deny De Rossi's comments.
He instead claimed Mandzukic had insulted his players throughout the game and said De Rossi's only mistake was not to cover his mouth with his hand, a technique regularly used by players and coaches to conceal what they are saying.