The 27-year-old, injured in training on Monday, will be replaced in the squad by fellow off-spinner Laura Marsh.
Hazell's place in the side for Tuesday's game with India went to left-arm spinner Rebecca Grundy.
"This is obviously very disappointing news for Danielle," said England head coach Mark Robinson.
"She was an important component of the attack, especially given her excellent current form and the conditions here in India, but we are fortunate to have such a talented and experienced replacement in Laura."
Durham-born Hazell is second in the International Cricket Council's rankings for bowlers in women's Twenty20 internationals, behind England pace bowler Anya Shrubsole.
Kent's Marsh, 29, has played 58 Twenty20 internationals for England, including the World Twenty20 success in 2009.
Although she was left out of England's squad for the recent series in South Africa, she has played T20 cricket for Sydney Sixers and Otago this winter.
She will join the squad in time for Sunday's game with Pakistan in Chennai.
England spinner Danielle Hazell has been ruled out of the rest of the Women's World Twenty20 with a calf muscle strain.