Video footage from inside the stadium shows a post-match brawl where a number of punches are exchanged between two men, leaving one bloodied and in need of medical attention.
Metropolitan Police officers arrested two people in Trafalgar Square, central London, where the rival fans congregated before the game.
At least one person was treated in an ambulance in the square, with another pictured bleeding from the leg.
Public toilets were closed at 4:30pm as the result of deliberate damage, seeing equipment kicked and tiles pulled off.
The police arrested one fan on suspicion of common assault and another on the suspicion of being drunk and disorderly.
A Met Police spokesperson said before the match: "We are aware that in the past Scottish supporters have congregated in Trafalgar Square.
"Given the match falls on November 11, we have also been liaising with the Royal British Legion, who will be hosting their 'Silence in the Square' event in Trafalgar Square to mark Armistice Day."
Fans left a large number of beer cans and shattered glass bottles in the square, along with a traffic cone on top of one of the iconic lions flanking Nelson's Column.
The square's two fountains had been switched off prior to festivities after Scottish fans filled it with fairy liquid three years ago.
Passer-by Nicoletta Morelli told reporters: "It's not so nice to see London like this. I have not forgotten the hooliganism of British fans in Rome."
The match, which England won 3-0, caused controversy as both teams and their coaching flouted a Fifa ruling banning them from wearing remembrance poppies and poppy armbands.
Additional reporting by PAReuse content