Tevez also expressed his regret after displaying the sign, which read 'RIP Fergie', during City's celebratory open-top bus parade through Manchester city centre.
The 28 year old is thought to have been passed or thrown the placard by a fan as the squad travelled the streets with the Premier League trophy.
A City spokeswoman said: "The creation of the tasteless material is in itself reprehensible and in accepting and brandishing it, Carlos has made a significant error of judgement.
"The club wishes to express its sincerest apologies to Sir Alex Ferguson and Manchester United Football Club for any offence or distress caused."
The sign is thought to have been a reference to a remark by Ferguson three years ago when asked if United would ever be underdogs against City. "Not in my lifetime," the Scot said.
Tevez, who spent two years at Old Trafford before leaving for City in 2009, said: "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."
City beat arch-rivals United to the title on goal difference after a dramatic last-gasp 3-2 win over QPR on the final day of the season.
More than 100,000 people lined the streets to celebrate the achievement, which was City's first league triumph in 44 years.
Members of the victorious squad and Roberto Mancini, the manager, addressed fans from a stage in Albert Square before boarding their bus to greet more crowds.
Mario Balotelli, the controversial striker, was the only notable absentee but he had been expected to report back to Italy for medical checks after being named in his country's provisional squad for Euro 2012.
Mancini said: "I think we should be proud of these players because they did everything for you [the fans] – they wanted to win this championship for you."
Midfielder James Milner said: "I've not slept much – I couldn't.
"I'm still in a bit of shock – we all are. It's still hard to believe we were 2-1 down going into injury time so it's going to take a while to sink in."
Teammate Gareth Barry said: "Days like this are why we came to City and it's just fantastic.
"I've got memories I'll treasure for the rest of my career."
Crowds started gathering for the evening parade from mid-afternoon.
They were treated to big-screen re-runs of the title-clinching goals and other memorable wins while club greats Mike Summerbee and Tony Book, title-winners in 1968, also spoke.
Summerbee quipped that he hoped Ferguson was listening before paying tribute to the fans.
He said: "The people out here think so much of the club, these are proper supporters. This is what Manchester City is like."
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