Born in Wilmington, North Carolina, Delmonico graduated from New Hanover High School in 1976 and Liberty University in 1980 with a bachelor's degree in physical education. He lettered in baseball and soccer. He holds a masters degree from Clemson University in education administration.
Delmonico was Assistant Coach at Gloucester County College in 1980, Graduate Assistant Coach at Clemson from 1981–83, and Assistant Coach at Florida State University from 1984-89.
Delmonico was head coach of the Tennessee Volunteers baseball team from 1990 until 2007, when he was released from his contract. He returned to Florida State as a volunteer assistant coach, then joined Florida International University as an assistant coach. He was named manager of the Netherlands national baseball team and managed the team in the 2009 World Baseball Classic.
Delmonico is divorced from Barbara Vanaman of Williamstown, New Jersey. They have three sons: Tony, Joey and Nicky. Tony played for his father at Tennessee and is currently a prospect in the Los Angeles Dodgers organization. Nicky plays minor league baseball in the Chicago White Sox organization.