Spain on Monday demanded a "public apology" from Argentina's President Javier Milei for calling Spanish Prime Minister Pedro Sanchez's wife "corrupt" amidst a deepening diplomatic crisis between the Hispanic allies.

The socialist government has already recalled its ambassador to Buenos Aires and Foreign Minister Jose Manuel Albares said on Cadena Ser Radio that he would summon the Argentine ambassador on Monday.

"I am going to explain to him the gravity of the situation and I am going to demand again a public apology by Javier Milei," Albares said.

Albares said he could not exclude the rupture of diplomatic ties with Argentina if no apology was made.

"We clearly do not want to take these measures but if there is no public apology, we will do it," the minister said.

The tub-thumping Argentina leader caused Spanish government outrage at a conference in Madrid organised by Spain's far-right Vox party.

Milei lashed out at socialism and attacked Sanchez's wife, Begona Gomez, without naming her.

"The global elites don't realise how destructive it can be to implement the ideas of socialism," Milei said.

"They don't know the type of society and country that can produce, the type of people clinging to power and the level of abuse that generates."   

He added: "When you have a corrupt wife, let's say, it gets dirty, and you take five days to think about it."

Spain's prime minister recently considered resigning after Spanish prosecutors opened a preliminary corruption investigation against his wife which was quickly closed.

Within hours of Milei's attack, Spain recalled its ambassador and Albares slammed the visiting president's "insult". The European Union's foreign affairs chief Josep Borrell, also condemned Milei's latest remarks.

Milei's government has been unrepentant. "There is no apology to make No apology," said Argentina's Interior Minister Guillermo Francos. "I think, on the contrary, it is for the Spanish government to make an apology for what has been said about Milei," he told the TN television channel.

Milei arrived in Spain on Friday and there was immediate diplomatic friction when no meetings with Sanchez or King Felipe VI were organised during his stay. During a speech on his first day in Spain, Milei denounced what he called "satanic" socialism.

Milei, a self-declared "anarcho-capitalist" won elections last November and took office vowing to reduce Argentina's vast public debt to zero.

He has instituted an austerity programme that has seen the government slash subsidies for transport, fuel and energy.

Milei was one of a host of European far-right and populist leaders to speak at the Madrid conference, personally or by video.

Marine Le Pen, France's far-right standard bearer, spoke at the event while Hungary's Prime Minister Viktor Urban and Italy's Prime Minister Giorgia Meloni sent video messages.