The 21-year-old Crane, who made his debut in the fifth Ashes test in Sydney in January, felt discomfort in his back earlier this week and had two scans, which revealed a partial stress fracture, the ECB said in a statement.
He will return home and be replaced by left-arm spinner Jack Leach.
Crane had figures of 1-193 on debut and with New Zealand pitches rarely favouring spin, probably would not have been in contention for either of the matches.
The opening game from March 22 is New Zealand's first day-night test, with the second match at Hagley Oval in Christchurch from March 30.
(Reporting by Greg Stutchbury; Editing by John O'Brien)
