The terrorist organisation has been present in the Egyptian Sinai since 2017.
"Suffocation, suffocation, all of them dead," a witness was quoted as saying by AP.
Tre crime is believed to have been caused by religious hatred.
ICC ties this attack to others that have taken place across Egypt in recent months.
Local authorities assigned a security force to protect the church, but it was overwhelmed by the sheer scale of the mob.
The Coptic Orthodox Diocese of Alexandria declared the murdered priest a martyr.
Violence like this against Christians in Egypt has increased recently.
A second person, Sherif Rashad, sustained a gunshot wound and was transferred to a hospital in Alexandria.