The right-hander was realistically in England’s squad for the four-day Test as a spare batter, but the injury is a cruel blow.
He was set to make his Test debut last November in New Zealand but broke a thumb in practice before the first Test.
Cox has been in good form for Essex this season, scoring 82 against Nottinghamshire and 117 against Surrey before retiring hurt while on 103 against Somerset.
Zimbabwe’s last Test in the UK was in 2003, a series famous for the Test debut of James Anderson, England’s all-time leading wicket-taker.
England’s five-Test series against India begins on 20 June at Headingley.
England squad: Ben Stokes (Durham) – captain, Gus Atkinson (Surrey), Shoaib Bashir (Somerset), Harry Brook (Yorkshire), Sam Cook (Essex), Zak Crawley (Kent), Ben Duckett (Nottinghamshire), Ollie Pope (Surrey), Matthew Potts (Durham), James Rew (Somerset), Joe Root (Yorkshire), Jamie Smith (Surrey), Josh Tongue (Nottinghamshire).
Highlights of all England’s home international cricket and live coverage of The Hundred will be across BBC TV, iPlayer and the BBC Sport website, beginning with England women’s T20 international against West Indies on 21 May.
