The government of the Falkland Islands said it was "disappointed" about the banner and that they hope Fifa will "sanction all behaviour of this nature in line with its own rules".
They also said "we do not wish to see politics being brought into sport".
Milei was reported by Argentine broadcaster TN as saying that it was "perfectly valid and right that the players should want to express themselves and do so".
However, he went on to say that this should not lead to "bad interpretations," adding that "It's a game of football, that's how the team manager and the veterans [in Argentina of the 1982 Falklands War] understood it".
Peter Kyle, the business and trade secretary, said it is "essential" that politics is kept out of the World Cup and called for Fifa to investigate.
"I'm really proud of our team, what they achieved, the professionalism they showed, but also the dignity that they showed," said Kyle.
"And that was in real contrast to what we saw with the Argentine team last night. When it comes to the consequences for their actions, well, that's for Fifa to do now. And I really do hope that Fifa do a proper investigation into it".
Kemi Badenoch, the leader of the Conservative Party, echoed Kyle's sentiments and said that Fifa "absolutely should investigate".