Mulgrew was sent off following the off-the-ball indicent
Rangers have fined Kyle Lafferty for feigning injury in an incident that led to Aberdeen's Charlie Mulgrew being sent off on Saturday.
The forward, who has made a public apology, fell clutching his face after he and Mulgrew had gone head to head during Rangers' 2-1 victory at Ibrox.
"It was something I should not have done and I deeply regret what happened," he told Rangers' website.
"I will make a personal apology to Charlie Mulgrew."
Lafferty says that the Aberdeen defender, whose club is to lodge an appeal to the Scottish FA, should not have been sent off.
It has since been announced that it was Stuart Dougal's last game as a top-flight referee as he has decided to retire a week early.
And Lafferty said: "I also want to apologise to the referee and his assistants.
There are standards at Rangers Football Club that need to be upheld and I have fallen way below these standards
Kyle Lafferty
"It is difficult enough for officials to handle important SPL matches without me making it more difficult by behaving the way I did."
Television replays showed that Lafferty fell to the ground in a dramatic manner after he and Mulgrew squared up to one another following the forward's challenge on the Aberdeen player.
The Scottish Football Association is likely to take action against the Northern Ireland international.
It will invoke the same Uefa rule that led to Lithuania's Saulius Mikoliunas being punished for diving in a Scotland game, but whatever action the SFA takes, it will not take effect until next season.
Lafferty will therefore still be available for Rangers' final league match against Dundee United at Tannadice.
In his statement, he also accepts that he has let down his club, whose 2-1 victory over Aberdeen put them two points ahead at the top of the Scottish Premier League with one game remaining.
"There are standards at Rangers Football Club that need to be upheld and I have fallen way below these standards," said Lafferty.
"What I did was wrong and I would like to say sorry to the supporters because they should expect better from a Rangers player."
Assistant manager at Ibrox Ally McCoist told Rangers World: "We spoke to Kyle at great length.
"It is obviously something that he deeply regrets. It was a foolish thing to do.
"It's hopefully something Kyle will learn from. He now knows he has made a mistake, it is not expected from a Rangers player.
"The only thing he could have done was apologise and he did, so we can move on."
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