Watson was born in Deep Gap, North Carolina. According to Watson on his three-CD biographical recording Legacy, he got the nickname "Doc" during a live radio broadcast when the announcer remarked that his given name Arthel was odd and he needed an easy nickname. A fan in the crowd shouted "Call him Doc!", presumably in reference to the literary character Sherlock Holmes's companion, Doctor Watson. The name stuck.
An eye infection caused Watson to lose his vision before his second birthday. He attended North Carolina's school for the blind, the Governor Morehead School, in Raleigh, North Carolina.
In a 1989 radio interview with Terry Gross on the Fresh Air show on National Public Radio, Watson explained how he got his first guitar. His father told him that if he and his brother David chopped down all the small dead chestnut trees along the edge of their field, they could sell the wood to a tannery. Watson bought a Sears Silvertone from Sears Roebuck with his earnings, while his brother bought a new suit. Later in that same interview, Watson explained that his first high-quality guitar was a Martin D-18.
Watson's earliest influences were country roots musicians and groups such as the Carter Family and Jimmie Rodgers. The first song he learned to play on the guitar was "When Roses Bloom in Dixieland", first recorded by the Carter Family in 1930. Watson stated in an interview with American Songwriter that, "Jimmie Rodgers was the first man that I started to claim as my favorite." Watson proved to be a natural musical talent and within months was performing on local street corners playing songs from the Delmore Brothers, Louvin Brothers, and Monroe Brothers alongside his brother Linny. By the time Watson reached adulthood, he had become a proficient acoustic and electric guitar player.