Welshman Flower was sent off scarcely two minutes into St Helens' 14-6 win at Old Trafford, after punching Saints stand off Lance Hohaia in the face and kneeling over him to strike him flush to the head again.
The incident put Hohaia out of the game and prompted a melee between players from both sides. Police were also reviewing the incident, British media reported.
"There is no defence for my actions," Flower said in a statement released by Wigan. "It was something that happened in the heat of the moment when our emotions and adrenaline was running high.
"I instantly regretted it and am devastated that I allowed myself to punch someone like this."
Wigan chairman and owner Ian Lenagan and head coach Shaun Wane also apologised for Flower's "totally inappropriate" behaviour.
"Rugby league is a tough sport, which we are proud of, but such a response is neither encouraged nor condoned," they said in a joint statement.
Hohaia said he held no grudges.
"In the heat of the moment, people do some things they regret. I've done some silly things on the field myself," the BBC quoted the New Zealander as saying.
"He's probably disappointed in himself for not being able to play a part in the game as well."
(Reporting by Ian Ransom in Melbourne; Editing by Peter Rutherford)
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