Laghi was named Nuncio to Argentina on 27 April 1974. His six years in Argentina coincided with the worst excesses of the military dictatorship there in what is now known as the "dirty war". Many in Argentina believe that the church hierarchy supported the generals in their misrule, and that Laghi – who played tennis regularly with one of the leaders of the junta, Admiral Emilio Eduardo Massera – turned a blind eye to the murder and disappearances of thousands.[1][2][3][4] "Perhaps I wasn't a hero", he later said of his time in Buenos Aires, "but I was no accomplice".[citation needed] The debate that his conduct generated was enough, however, to ruin any chance he ever had of becoming pope.[citation needed] In his book In the Closet of the Vatican: Power, Homosexuality and Hypocrisy, Frédéric Martel alleges that Laghi was blackmailed by the Argentinean junta into not opposing them because of his use of so-called "taxiboys".[5]
Pope John Paul II transferred him as Apostolic Delegate (10 December 1980) and later (26 March 1984) Pro-Nuncio to the United States. In that role he helped place conservatives in key positions, such as Bernard Francis Law in Boston and John Joseph O'Connor in New York City.
On 26 December 1994, the American magazine Time published an article titled "Who Will Be First Among Us?" which prematurely speculated on who would be the next pope "as John Paul approaches the twilight of his papacy." Laghi was listed among eight cardinals who were considered leading candidates; the article identified Laghi as head of the Congregation for Catholic Education with conservative credentials. In 1997 the Dirty War scandal was widely exposed, and, though Laghi denied the charges, it was considered to have ended his chance for the papacy.
He headed the Congregation for Catholic Education until his resignation on 15 November 1999. From 1999 to 2001 he was the Cardinal Protodeacon (the longest serving Cardinal Deacon), before exercising his right as a Cardinal Deacon of ten years' standing to become a Cardinal-Priest of San Pietro in Vincoli. On his 80th birthday, 21 May 2002, he became ineligible to vote in a papal conclave but continued to undertake special missions for his friend John Paul II.
In 2000, Laghi was awarded the F. Sadlier Dinger Award in recognition of his outstanding contributions to the ministry of religious education in America.
On 1 March 2003, Laghi, as special papal envoy to the United States, met with President George W. Bush and conveyed the Pope's request that the United States reconsider the decision to go to war against Iraq. Bush was photographed with Laghi and commented that he was "an old family friend". During the presidency of Bush's father, George H. W. Bush, Laghi, as Nuncio to the United States, was a frequent guest of Bush senior and his family.
Laghi died in Rome on 10 January 2009, from a blood disorder, aged 86.[7]