Gallinari, 62, died suddenly, Italy's ANSA news agency reported.
He was one of the Red Brigades members known to have guarded former Italian Premier Aldo Moro while he was being held captive for 55 days before he was killed in 1978.
Moro was captured on March 16, 1978, by Red Brigades members as he headed for a confidence vote for a new government led by a Christian Democrat party candidate.
The Red Brigades proposed to exchange Moro for the release of several imprisoned terrorists but the Italian government refused to negotiate. After near two months of holding Moro hostage, the group fatally shot him inside a car.
It was long thought that Gallinari was the trigger man in Moro's shooting; however, the group's leader, Mario Moretti, confessed.