The Uruguayan had until Friday lunchtime to contest the Football Association's decision to add seven gamers to the usual three for a violent conduct charge but opted against it.
"I hope that the people I offended last Sunday will grant me forgiveness and I again repeat my personal apology to Branislav," Suarez said on his Twitter account.
"While 10 games is clearly greater than those bans given in past cases where players have actually been seriously injured, I acknowledge that my actions were not acceptable on the football pitch so I do not want to give the wrong impression to people by making an appeal."
Suarez, who bit Ivanovic on the arm during the 2-2 draw at Anfield last weekend, will not be eligible to play for Liverpool in domestic competition until September.
The FA charged Suarez with violent conduct and its Independent Regulatory Commission decided on Wednesday that the automatic three-game ban for a violent conduct charge was not sufficient, adding a further seven matches to his suspension.
Liverpool, who fined Suarez, reiterated their disappointment with the FA's decision to ban Suarez for 10 games.
"The charges against Luis were his to consider and we have to respect his decision to not appeal the 10-game ban," managing director Ian Ayre said.
"We are all disappointed at the severity of the punishment and, in particular, the differing standards that have been applied across various previous incidents.
"Luis is an important member of our team and nothing has changed in that regard.
"We are committed to helping him improve his conduct and he will be given our full support. We look forward to him returning to the team next season when he is available for selection."
Liverpool manager Brendan Rodgers has also criticised the FA's tough stance and there was some sympathy from rival managers on Friday.
"In this case, the proportionality looks very severe considering what other players have been punished for," Arsenal manager Arsene Wenger told reporters.
"I think what has gone completely against Suarez is his history, that is very heavy on the offence front.
"That's why he has been punished so severely, that's the only explanation I can find."
Manchester City's Roberto Mancini told a news conference: "Five or six games was enough, this is my opinion but I don't work for the FA."
It is the second time the 26-year-old Suarez has been punished for biting an opponent after he bit PSV Eindhoven's Otman Bakkal on the neck while playing for Ajax in 2010, earning a seven-match ban.
Suarez was also banned for eight games last season after being found guilty by the FA of racially abusing Manchester United's Patrice Evra in October 2011.
