The 25-year-old was held by police in Hertfordshire, north-west of London, on Monday in connection with damage to a building where he lives in the Bushey area.
Bendtner, writing on his Instagram account on Tuesday, said he'd accidentally damaged a door he tried to force open after it became stuck. "On Saturday night after our win against Southampton I went out with my family and friends for dinner," he wrote.
"After our dinner we all wanted a night swim in my gym so we changed to shorts and brought towels down. "As we arrived at the gym my key fob changed the colour from red (locked) to green (open) but the door remained closed.
"We thought the door was stuck so we tried to open it and it caused some damage. I apologise for any inconvenience for my neighbours and I have paid in full.
"We had no part in damaging anything else in the building. The matter is closed with the police, and my focus is on tonight's (Tuesday's) match (against Marseille in the Champions League)."
Police said Bendtner had been cautioned -- meaning he will not be required to attend court. A Hertfordshire Police statement said: "A 25-year-old man from Bushey has been cautioned for criminal damage and is no longer in custody.
"He was arrested in connection with an incident which occurred at an address in The Avenue, Bushey, in the early hours of November 24."
Denmark international Bendtner has scored 45 goals for the Gunners in 164 appearances since making his debut for the club in the 2005/06 season but is currently struggling for regular first team football with the Premier League leaders.
In June last year, Bendtner was handed a one-match ban and fined 100,000 euros by UEFA for lowering his shorts after scoring twice in his country's European Championship defeat by Portugal.
He was found guilty of improper conduct for his reaction to his second goal in the 3-2 Group B defeat, when he exposed underwear bearing a bookmaker's logo.
An Arsenal spokesman said Tuesday: "As a club we do not comment on players' private matters. However, we do expect the highest standards to be maintained on and off the pitch and will be reminding Nicklas of this."