"I am deeply sorry for my inexcusable behaviour earlier today during our match against Chelsea," Uruguayan Suarez, who later scored in the 97th minute to grab a 2-2 draw for Liverpool, said on the club's website.
"I have issued an apology and have tried to contact Branislav Ivanovic to speak to him personally. I apologise also to my manager, playing colleagues and everyone at Liverpool Football Club for letting them down."
It was an eventful afternoon for Suarez, who also apologised on Twitter.
He had set up Daniel Sturridge to make it 1-1 before handballing in the penalty area to concede a spot-kick from which Eden Hazard put Chelsea back in front.
The incident with Ivanovic occurred in the second half and followed a brief tussle in the penalty area between the two players.
"Luis has made an unreserved apology for his actions today," Liverpool managing director Ian Ayre said.
"His behaviour is not befitting of any player wearing a Liverpool shirt and Luis is aware that he has let himself and everyone associated with the club down. We will deal with the matter internally and await any action from the FA."
Manager Brendan Rodgers added: "Having reviewed the video footage and spoken to Luis, his behaviour is unacceptable and I have made him aware of this."
Suarez - who has repeatedly courted controversy during his time at Liverpool, most notably when being hit with an eight-match ban for racially abusing Patrice Evra - also came under fire from Professional Footballers' Association chief executive Gordon Taylor.
"It was not good to see to say the least. It was unacceptable," Taylor told Sky Sports News. "His reputation for controversy continues to increase when he can be such a good player, one of the world's best, so it is a real dilemma now.
"It's uncalled for, it's not good, it's depressing, it's such a shame for a player who's a really good footballer but this is not the first time that such things have happened. I know it's a really difficult one for Liverpool and the FA are going to have to deal with it."
The FA is expected to review the incident on Monday morning and decide whether to charge the striker, who has been named on the six-man shortlist for the PFA Player of the Year award.