Galindo began skating with his sister. Although the sport was expensive, his parents were supportive and forwent a chance to buy a house, settling instead for a larger trailer. As a singles career, Galindo won the 1987 World Junior title.
Galindo was paired with Kristi Yamaguchi by his coach, Jim Hulick. They placed 5th on the junior level at the 1985 U.S. Championships and won the junior title in 1986. Hulick died of AIDS-related cancer in 1989. Galindo did not compete in singles in the 1988-89 and 1989-90 seasons in order to concentrate on pairs. Galindo and Yamaguchi won the 1988 World Junior title and the U.S. senior championships in 1989 and 1990. However, in April 1990, their partnership came to an end when Yamaguchi decided to focus on her singles career. As there was no one of her caliber available, Galindo returned to singles competition.
Rudy Galindo's father died of a heart attack in 1993; his brother, George, died from AIDS in 1994, as did another coach, Rick Inglesi, in 1995. Galindo took eight months off after the 1995 U.S. Championships. However, with the following year's event in his hometown, presenting a chance to compete in front of his mother who no longer traveled, he decided to resume training in September 1995. In January 1996, he won the men's title at the U.S. Championships at the San Jose Arena, becoming the oldest male to win this title in 70 years. He went on to win a bronze medal at the 1996 World Championships. His sister, Laura, was his coach.
Galindo retired from eligible competition in the summer of 1996 and toured with Tom Collins' Champions on Ice. He underwent hip replacement surgery in August 2003 after finishing the season's tour with a broken femur on his left side. After recovering, Galindo continued to tour with COI until it went out of business in 2007. In 2006, Galindo was a judge on the WE tv series Skating's Next Star, created and produced by Major League Figure Skating and hosted by Kristi Yamaguchi.
Galindo coaches at Sharks Ice San Jose (Logitech Ice), the same rink where he trained during his competitive career. Among his students is Yamaguchi's daughter, Emma Hedican.
Galindo was inducted into the San Jose Sports Hall of Fame in 2011. He was elected to the U.S. Figure Skating Hall of Fame in December 2012.
Galindo is the third child of Jess and Margaret Galindo, having a brother, George, who was ten years older and sister, Laura, five years older. He is of Mexican descent by way of his grandparents on his father's side. In 1996 he came out as gay in Christine Brennan's book Inside Edge: A Revealing Journey Into the Secret World of Figure Skating (ISBN 0-385-48607-3), which was published shortly before he won his national title that year. He is the first openly gay skating champion in the U.S. His autobiography Icebreaker (ISBN 0-671-00390-9), co-written with Eric Marcus, was published in 1997.