Born in Buenos Aires, Argentina, Julio Bocca began his dance training with his mother at the age of four. He studied at the Instituto Superior de Arte del Teatro Colon where his teachers included Gloria Kazda, Jose Pares, Ninel Jultyeva, Karemina Moreno and Luis Aguilar. During that time, he danced with the Caracas Ballet Company performing the title role in Le Spectre de la Rose, among others. In 1983, he joined the Ballet del Teatro Municipal de Rio de Janeiro as a principal dancer and, in the same year, appeared with the ballet company at the Colon Theatre and with the International Ballet of Caracas.

In May 1985, he won the Gold Medal at the Fifth International Ballet Competition in Moscow. He has toured the former U.S.S.R., appearing with the ballet company in Novosibirsk, where he danced the Prince in The Nutcracker and Basil in Don Quixote, and with the Bolshoi Ballet in Moscow and the Leningrad-Kirov Ballet, again dancing in Don Quixote.

Bocca joined American Ballet Theatre as a Principal Dancer in 1986. His roles with the Company include the title role in Apollo, Solor in La Bayadere, a Stepsister in Cinderella, the Duet from Concerto Six Twenty-Two, Conrad in Le Corsaire, Basilio in Don Quixote, Albrecht and Hilarion in Giselle, Des Grieux in Manon, Danilo in The Merry Widow, the pas de deux Other Dances, Arthur Saint Leon in Pas des Deesses, Romeo in Romeo and Juliet, Prince Desire in The Sleeping Beauty, Misgir in The Snow Maiden, Prince Siegfried in Swan Lake, James in La Sylphide, Petruchio in The Taming of the Shrew, the Tchaikovsky Pas de Deux, the fourth variation in Variations for Four and leading roles in Drink To Me Only With Thine Eyes, Etudes and Theme and Variations. Bocca created the leading role in Brief Fling.

Bocca made his Broadway debut in Fosse.