Born in San Gabriel, Jalisco, Galindo studied intermittently from 1931 to 1944 at the National Conservatory in Mexico City, studying with Carlos Chávez (composition), Candelario Huizar, José Rolón, and Manuel Rodríguez Vizcarra (piano).
In 1934, he formed the Grupo de los cuatro with fellow composers Daniel Ayala, Salvador Contreras, and José Pablo Moncayo, seeking to use indigenous Mexican musical materials in art-music compositions (Stevenson 2001).
To do this, he would frequently "retreat from Mexico City to a house in the country for weeks at a time in order to immerse himself fully."
Conant wrote of Galindo, "He would, however, like to write an opera or even a cycle of operas dealing with the life of Mexico, the agitated life of Mexico from the Aztec leader Cuauhtémoc through the leaders of Mexican independence like Morelos, Juárez, Zapata, and Cardenas" (Conant 1977, [page needed]).
"José Angel Lamas" prize from Secretary of Public Education; honor diploma from the Venezuelan Society of Authors and Composers; gold medal as "the best composer of the year"; diploma from the Municipality of Los Angeles, California; National Arts and Science Award for 1958–1964, presented by the President of the republic of Mexico, Adolfo Lopez Mateos Galindo's compositions number over 150, and include works in a variety of styles and ensemble forces.