Médici started his career playing for San Telmo and then moved to Club Atlético Atlanta. In 1922 he joined Boca Juniors and made his first appearance for the Argentina national team. During his time with the club, Boca Juniors won four league titles and a number of minor cup competitions.
Médici appeared in five Copa América squads for Argentina between 1922 and 1926. He was part of the winning team in 1925.