Born in Pittsburgh, Pennsylvania, in 1921, Betz was the oldest of four children—two sons and two daughters—of Carl W. and Mary Leona Betz. His father was a native of Missouri; and according to the federal census of 1930, he was then the "chief chemist" at a local laboratory and later, by 1940, for Allegheny County. Growing up in the Pittsburgh suburbs of Crafton and Mt. Lebanon, Betz began his amateur acting career very early, at the age of 10, when he formed a theatrical company with six friends who performed plays in his grandmother's basement. After graduating from Mount Lebanon High School in 1939, he won a scholarship to Duquesne University. During the summer, Betz performed in a Pittsburgh summer stock company and decided to transfer to Carnegie Mellon University, then known as Carnegie Tech, in Pittsburgh to study drama. His education was interrupted in 1942, when he was drafted into the United States Army. He served in Italy and North Africa during World War II and eventually became a technical sergeant with the Corps of Engineers.
Betz was married twice and had one child. In June 1952 he married Actress Lois Harmon with whom he had a son, Richard. The couple separated in May 1960 and divorced in 1961. In December 1963, Betz married Gloria Stone Martin, sister of Actress Nita Talbot. Through his marriage to Martin, Betz had a stepdaughter, Rio. He and Gloria remained together for 15 years, until Carl's death.
In 1958, Betz was cast as pediatrician Dr. Alex Stone in ABC sitcom The Donna Reed Show. The show revolves around the home and school problems of a middle-class American family in the late 1950s and early 1960s. Alex was often called upon to rescue wife Donna Stone (Reed) from awkward situations and to monitor the behavior of their children, Mary (Shelley Fabares) and Jeff (Paul Petersen). Jeff Stone introduced the sentimental hit song "My Dad" in a 1962 episode, specifically singing the tune to Betz. The series was a hit for ABC and aired for eight seasons from September 1958 to March 1966. During the run of the series, Betz continued acting in stage roles during the show's hiatus. In 1964, he appeared as Reverend T. Lawrence Shannon in a limited stage run of The Night of the Iguana, for which he earned excellent reviews.
After Donna Reed's cancellation, Betz returned to television roles and stage work. In 1967, Producer Paul Monash offered Betz the role of defense Lawyer Clinton Judd in the legal drama Judd, for the Defense. Monash had seen Betz's performance in Night of the Iguana in 1964 and was impressed with his acting. Betz initially thought the role was for a guest spot but soon realized Monash had proposed that he star in a new series. Betz initially had misgivings, stating "I did not want to do another series, you get bored", but eventually relented because he liked the scripts. The series, which premiered on ABC in September 1967, was praised by critics but struggled in the ratings. Shortly after ABC canceled the series in 1969, Betz won the Primetime Emmy Award for Outstanding Continued Performance by an Actor in a Leading Role in a Dramatic Series for his work on the series. He also made many guest appearances on a variety of popular television shows such as Mission: Impossible, The Mod Squad, Love, American Style, and Starsky and Hutch. One of his last roles was as General Douglas MacArthur in the one-man stage play I Shall Return.
In 1977, Betz was diagnosed with terminal and inoperable lung cancer. He kept his diagnosis private so he could keep working. On November 29, 1977, Betz entered Cedars-Sinai Medical Center in Los Angeles. He died seven weeks later, on January 18, 1978. His remains were cremated.