The 16-year-old is alleged to have targeted Arbroath Abbey stonework with spray paint.
Repairs have been estimated to cost “well into” a four-figure sum.
Inspector Gary Aitken said: “This building is historic to the town of Arbroath and there is nothing to be achieved by damaging the Abbey other than creating needless expense and causing upset.
“Vandalism will not be tolerated and where offences are reported to Police Scotland, offenders will be dealt with appropriately.”
Officers made an appeal about the incident last week, saying it had been targeted over two nights between 5pm on Tuesday, July 3 and 7am on Thursday July 5.
The A-listed building in Angus was founded in 1178 by King William the Lion for a group of monks.