Security officials, speaking on condition of anonymity, said that a third soldier appeared to have carried out Saturday's shooting, near the Taba border crossing.
The officials identified the deceased as Ihab Gouda Mohammed, 22, and the wounded soldier as Mohammed Diyab Hassan. The two were taken to Mubarak Hospital in El Arish, they said.
The third soldier suspected of being behind the shooting was being questioned, they said.
Shootings are common in the Sinai, but violence among Egyptian soldiers is rare.
Sinai was occupied by Israel during the 1967 Six Day but was returned to Egypt in 1982 after the countries signed a peace treaty.