Tevez held a sign with the text "R.I.P. Fergie" high over his head while aboard City's open-top bus, which drove through the streets of Manchester as about 100,000 fans came out to celebrate.
"The creation of the tasteless material is in itself reprehensible and in accepting and brandishing it, Carlos has made a significant error of judgment," City said in a statement. "The club wishes to express its sincerest apologies to Sir Alex Ferguson and Manchester United Football Club for any offence or distress caused."
Tevez left United to make the rare move across Manchester to City in 2009 after being unhappy at his lack of playing time under Ferguson.
"I got carried away in the excitement of the moment and I certainly didn't mean any disrespect to Sir Alex Ferguson, who I admire as a man and a manager," Tevez said.
The incident comes at the end of a season full of turmoil for Tevez, who spent several months in Argentina during a dispute with City which cost him more than $15 million in fines and lost earnings.
City beat United to the title on goal difference on Sunday after scoring twice in stoppage time to beat Queens Park Rangers 3-2.