Díaz was born in San Juan, Puerto Rico, the capital of the island, but lived and was raised in the town of Cataño.

After finishing his primary education, Díaz attended the University of Puerto Rico High School in Río Piedras.

After the tour, he entered and participated in a competition celebrated at the Metropolitan Opera House, winning third place in the New England region.

On March 29, 1963, Díaz won the Metropolitan Opera National Council Auditions, becoming the first Puerto Rican to obtain such an honor.

In 1966 he helped to inaugurate the Lincoln Center in New York City by starring opposite Leontyne Price in the opening night performance of Antony and Cleopatra by Samuel Barber.

In 1986, Díaz played the role of Iago, opposite Plácido Domingo in Franco Zeffirelli's film adaptation of Otello.

In 1967, his Escamillo in Carmen was filmed, conducted and directed by Herbert von Karajan, and also starring Grace Bumbry and Jon Vickers.

He has also sung the role of Emile de Becque in the world-premiere recording of the complete score from Rodgers and Hammerstein's South Pacific.