Galindez aspired to become both a boxer and a Formula 1 driver since he was little, but he was more interested in boxing. As an amateur, he compiled a winning record and participated in the 1968 Olympic games of Mexico City, where he lost in a preliminary round bout.
Galindez then turned professional, and on May 10 of 1969, he debuted as a paid fighter with a win over Ramon Ruiz by a knockout in four at Buenos Aires. After one more win, he faced Adolfo Cejas in Azul, Argentina, in a fight which resulted in a ten round draw.
In 1970, Galindez had 10 fights, span during which he went 5-3-1, with one no contest. He suffered his first loss, at the hands of Juan Aguilar, by a decision in ten, and had a second and third fight with Aguilar, of which the second ended in a first round no contest, and the second in another Aguilar decision win. He began his five fight rivalty with Jorge Ahumada knocking Ahumada out in five rounds, and lost for the Argentinian Light Heavyweight title with a twelve round decision defeat against Avenamar Peralta.
He and Peralta had a immediate, non-title rematch in 1971, and Galindez won by a ten round decision. '71 was actually a year of rematches for Galindez, the only new boxer that he met that year being Pedro Rimovsky, and he too, had a rematch with Galindez before that year was over. He lost to Ahumada in the second of the five fights between them, by a decision at Mendoza, Argentina, then beat him in fight three by a knockout in nine , and in fight four by a knockout in 6, both the third and fourth fight being held at Buenos Aires. He and Rimovsky had a first round no contest in their first bout, and drew over the ten round distance in the rematch, and then Galindez had two more fights with Peralta, losing by a knockout in nine and a decision in 10.
In 1973, Galindez had eight fights, winning seven and drawing one. He was finally able to obtain Argentina's Light Heavyweight title, by beating Aguilar by a decision in twelve. He beat Aguilar once again, by a knockout in six, and he also beat Eddie Owens, Eddie Duncan and Raul Loyola, the latter being beaten twice, one time defending his Argentinian title.
He began 1974 with a step up in class, meeting former world title challenger Ray Anderson, beating him by a knockout in two. After seven wins in a row, he was given his first shot at a world title, facing Len Hutchings for the WBA's vacant world Light Heavyweight championship. Galindez then joined Torres and Rondon as the only Hispanic Light Heavyweight world champions in history, and Carlos Monzon as the only Argentinian world champions of that era, when he knocked out Hutchings in the thirteenth round to become the WBA's world Light Heavyweight champion.
Ironically, Galindez, who never fought outside Argentina before becoming a world champion, became a travelling fighter after that. Once he beat John Griffin in Buenos Aires by knockout in six in a non title bout, it was off to South Africa, for his first international fight, which was, at the same time, his first world title defense. He beat Pierre Fourie there by a decision. Next was Las Vegas, for his first fight in the United States, another non title affair. He knocked out Ray Elson in eight there. Next stop was the Madison Square Garden in New York, where he and arch-rival Ahumada met for a fifth time, this time with the world title on the line. After beating Ahumada by a decision in 15, Galindez returned to South Africa, where he once again beat Fourie by decision in 15.
1976 saw a trip to Norway, where he beat Harald Skag by a knockout in three to retain the title, and to Denmark, where he beat Jesse Burnett by a decision in ten in a non tite affair. Then came a third trip to South Africa, where he knocked out challenger Richie Kates in the 15th and last round, and, after a fight with Billy Douglas in Buenos Aires, (a win by decision in ten) he found himself in South Africa once again, where he beat Kosie Smith by a decision in fifteen to once again, retain the title.
After beginning 1977 by beating Guillermo Aquirrezabala by a knockout in four in Mendoza, Argentina, he and Kates had a rematch in Rome, Italy, where Galindez once again beat Kates by decision over 15. Next, the Argentinian champion and challenger Alvaro Yaqui Lopez boxed a 15 round title bout, once again in Italy, and Galindez retained the title by a decision in Rome. Galindez' last fight of '77 took him to Italy for a third time, and he beat future world champion Eddie Mustapha Muhammad (then Eddie Gregory) by a decision in fifteen.
In 1978, Galindez went to Italy again, retaining the title with a fifteen round decision in a rematch with Lopez, and then he made a series of non title bouts in Argentina before he went on the road again, this time losing his title to Mike Rossman by a knockout in 13, in the same undercard where Muhammad Ali recovered the world Heavyweight title for the third time by beating Leon Spinks at the Superdome in New Orleans.
After boxing his first fight in 1979, beating Roberto Aguilar by a knockout in six back home, he and Rossman were supposed to have a rematch in February of that year, but Galindez refused to fight, arguing that the judges selected for the rematch would probably favor Rossman. However, the rematch did come off later that year, and Galindez recovered the world championship, once again at New Orleans, knocking Rossman out in 10 rounds. This time, however, he wouldn't last long as king of the Light Heavyweights, and he lost the title in his first defense, by a knockout in 11 to Marvin Johnson, a boxer who'd later join Muhammad Ali, Sugar Ray Robinson and Carlos De Leon among others as one of the few boxers to be world champions three times in the same category. The fight with Johnson took place in New Orleans too.
After losing a rematch with Burnett in 1980 by a decision in twelve in Oakland, California, Galindez was forced to retire because of two operations to repair his detached retinas, and then he tried to pursue his other dream of becoming a formula one driving champion. Once he reached the formula one premier league in Argentina, however, his life was cut short by tragedy: On October 25 of that year, he participated in what would be his first and last formula one race. After going to the pit for car repairs, Galindez was struck by a car that lost control and killed instantly while on the waiting area.
He had a boxing record of 52 wins, 9 losses and 4 draws with 2 no contests, and 34 wins by knockout.
He is a member of the International Boxing Hall Of Fame.