The defending champions suffered a 1-0 extra-time loss in New Jersey after Ferran Torres struck in the 106th minute.
Argentina midfielder Enzo Fernandez was sent off just before the additional period for a second yellow card, while there were unsavoury scenes at full-time involving Leandro Paredes and Nahuel Molina.
BBC pundit Hart said: "There was one man on the pitch who did have class and that was Lionel Messi, who shook hands with every one of those Spanish players.
"The game's done. They could have absolutely no complaints about what happened today. Disgusting behaviour."
Hart added: "Spain fully deserve it. They have been absolutely superb all tournament.
"They have dominated all game against a rigid, rugged Argentina. The best team has won this World Cup."
Argentina were previously criticised for their aggressive behaviour during their 2-1 semi-final win over England, after which players unveiled a banner claiming sovereignty of the Falklands Islands.
Fellow pundit Wayne Rooney said of Argentina's behaviour: "Does it surprise you? It doesn't surprise me.
"We've seen this with Argentina before. It's just not the reaction you want. We've seen some of the reaction after knocking England out in the last round.
"That's just sad. If you lose a game of football, be graceful and get out of there. It's really poor from them."
Argentina defender Molina struck Spain midfielder Rodri as he ran on to the pitch to celebrate his side's success.
Paredes appeared to grab Eric Garcia by the throat and then threw Gavi to the ground.
"That's a reaction to them being outplayed, out-fought, out-thought on the football pitch, and then they resort to that at the end," continued Rooney.
"It's poor from those two players."
Gary Neville, speaking on ITV, singled out Paredes' behaviour for criticism.
The former England full-back said: "They're very competitive and I love that element of them, but (Paredes) has just been a disgrace for the last couple of games, and why the referees have allowed him to continue doing what he is doing."
Micah Richards added: "It's quite embarrassing really. In the game, they were just on the cusp every single time.
"Paredes doesn't need to do that, he's better than that, and it doesn't stop there – it keeps going and going and going.
"We all know what it's like to lose a game, we can be very frustrated, but this is not good enough – it's not a good look at all."
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