Welch studied history in Abilene, Texas, at Abilene Christian College, now Abilene Christian University, where he was a varsity cheerleader and a member of Phi Delta Psi social club. In 1940, he received his Bachelor of Arts in history. He was married on December 17, 1940 to his first wife, Iola Faye Cure, in a ceremony performed by Homer Hailey, a professor at Abilene Christian .

Welch served as the Mayor of Houston, Texas from 1964 to 1973. Welch was known for his quick West Texas quips which made clear to the media he would stand up for what he believed was right.

Houston grew immensely when Welch was mayor. In 1963, Houston's population reached over one million people, yet was then still considered a "small" city in the eyes of the national media. Under Welch, several events put Houston prominently on the U.S. and world maps, including the opening of the Astrodome in 1965 and the Houston Intercontinental Airport in 1969. NASA at nearby Mission Control sent a man to the moon. Lake Conroe and Lake Livingston opened to provide water for Houston. Welch also closed forty inefficient sewage treatment plants, began cleaning up the Houston Ship Channel, focused on bayou beautification; and began development of the downtown Civic Center, among other accomplishments. Welch was the first Houston mayor to win all precincts during one of his reelections, including predominantly African American areas with which some claim he had trouble.

By the time that he  left office in 1973, Houston was within two years of supplanting Detroit to become the fifth largest city in the United States in 1975, and in the fall of 1980, the fourth.

Louie Welch died of lung cancer in January 27, 2008 in Harris County, Texas