Gwynn, who played his entire career with the San Diego Padres, died early Monday morning at Pomerado Hospital in Poway, California, surrounded by his family, the Cooperstown museum said in a statement.
Gwynn, who had two operations for cancer in his right cheek, had signed an extension as baseball coach at San Diego State earlier this month after having been on medical leave since late March while recovering from cancer treatment. He took over the program at his alma mater after the 2002 season.
The sweet-swinging lefthanded hitting outfielder ammassed 3,141 hits over 20 seasons, compiling a career batting average of .338 that is 18th best all-time and claimed eight National League batting titles.
A 15-time All-Star, Gwynn also won five Gold Glove Awards in recognition of his defensive skills and was elected to the Hall of Fame in 2007 in his first year of eligibility.
Gwynn was a key member of the 1984 and 1998 San Diego Padres National League pennant-winning teams that reached the World Series and he batted .371 in the Fall Classic.
"It is with profound sadness that we mourn the passing of Tony Gwynn,” said Jane Forbes Clark, Chairman of the Board of the National Baseball Hall of Fame and Museum.
“He was beloved by so many, especially the Hall of Fame family, for his kindness, graciousness and passion for the game. Tony was one of baseball history’s most consistent hitters and most affable personalities."
Gwynn's Hall of Fame plaque called him "an artisan with the bat".
Major League Baseball commissioner Bud Selig said in a statement: “Major League Baseball today mourns the tragic loss of Tony Gwynn, the greatest Padre ever and one of the most accomplished hitters that our game has ever known, whose all-around excellence on the field was surpassed by his exuberant personality and genial disposition in life.
"Tony loved our game, the city of San Diego and his alma mater where he starred and coached, San Diego State University, and he was a part of a wonderful baseball family."
Gwynn's No. 19 was retired by the Padres in 2004.