Crawley, 26, picked up the problem during England's third Test win over the West Indies last month. Seamer Dillon Pennington will also be out with a hamstring injury.
Middle-order batter Jordan Cox has received a first call-up to the Test squad. The 23-year-old had previously toured Pakistan with the white-ball squad in 2022.
Seamer Olly Stone will also return to the Test squad for the first time since June 2021.
Captained by Ben Stokes, England will host Sri Lanka at Old Trafford for the first Test starting August 21.
England Test squad:
Ben Stokes (captain), Gus Atkinson, Shoaib Bashir, Harry Brook, Jordan Cox, Ben Duckett, Dan Lawrence, Ollie Pope, Matthew Potts, Joe Root, Jamie Smith, Olly Stone, Chris Woakes and Mark Wood.