Police told the BBC Charlie Gilmour, 21, tried to damage the flag on the Cenotaph war memorial Thursday and was arrested on suspicion of violent disorder and attempted criminal damage.
Gilmour, a student at Cambridge, apologized Friday, saying he was "mortified by his moment of idiocy."
He added, "I would like to express my deepest apologies for the terrible insult to the thousands of people who died bravely for our country that my actions represented."
The rock star's adopted son was the 35th person to be held after violent demonstrations against tuition fee increases. Photos of him swinging from a flag appeared on many front pages.
On Saturday night, the police have issued video images of 14 alleged rioters and asked the public to help identify them.
A total of 176 people were arrested during the demonstrations, including 34 who were detained Thursday and Gilmour.