The 25-year-old was arrested and questioned by police on Monday following the incident, which occurred just hours after the Gunners had beaten Southampton 2-0 at the Emirates Stadium on Saturday.
Hertfordshire police has since confirmed that the Denmark international has been released from custody and no further action will be taken.
But while his club will not be making any public statement about Bendtner's antics, they have confirmed that they will remind the Dane that they "expect the highest standards to be maintained on and off the pitch".
Bendtner has become Arsenal's second-choice striker this season despite missing out on a summer move to Crystal Palace because he was nine pounds overweight.
He has recently confirmed, however, that he wishes to leave the club as soon as possible, and joked that he hopes Barcelona or Real Madrid are his next destination.
Earlier this year a Copenhagen court banned Bendtner from driving for three years and fined him nearly £100,000 for drink driving. He was also banned from playing for Denmark for six months over the incident, but has since returned to the international fray.