Fathers 4 Justice, a protest group that campaigns on behalf of fathers denied contact with their children, said the arrested man was a member. It said he had written "Help" with paint on the picture in the abbey's Chapter House.
Police said a 41-year-old man was arrested on suspicion of criminal damage and taken to a London police station.
The abbey said the painting by Ralph Heimans was vandalized at lunchtime on Thursday. It has been removed from public view until it can be restored.
The portrait by Australia-born artist Heimans was commissioned to mark last year's anniversary of the queen's 60 years on the throne. It was on display until March in Australia's National Portrait Gallery in Canberra.
Last week, the monarch attended a ceremony in the abbey to mark the 60th anniversary of her coronation there in June 1953.