Argentina beat 2018 winners France on penalties to lift the trophy for a third time.
Goalkeeper Emiliano Martinez made a lewd gesture with his goalkeeper of the tournament trophy after the match, before being filmed in the dressing room mocking French star Kylian Mbappe.
Mbappe won the FIFA Golden Boot trophy after scoring the most goals in the tournament.
Football’s world governing body said Argentina had potentially breached rules regarding “offensive behaviour and violations of the principles of fair play” and “misconduct of players and officials”.
FIFA is also investigating the Argentinian FA over breaching media and marketing regulations.
Martinez has also come under criticism from pundits and social media for mocking Mbappe during Argentina’s victory parade in Buenos Aires – where he held a baby doll with the French player’s face plastered on it.
The Croatian team is also being investigated for potential violations of “discrimination” and “order and security at matches” rules.
Players appeared to sing a song with fascist connotations following their third-place playoff victory over Morocco.
Defender Dejan Lovren defended himself and his teammates, recently telling AFP news agency it was “a patriotic song which relates to my country, which means that I love my country”.
Meanwhile, FIFA said it had fined Mexico 100,000 Swiss francs ($107,840) for chants by its supporters during the team’s game against Poland and Saudi Arabia during the group stages.
The team will also be forced to play one FIFA competition match behind closed doors.