The murderer was nicknamed "the chessboard killer" by the media because he told detectives in a confession that he had hoped to put a coin on every square of a chessboard for each of his victims.
Hamas loyalists claimed that Odai al-Rabei's alleged participation in anti-Hamas protests meant he was a "traitor."
The 84-year-old's remains were discovered at his home on May 13 with a number of suspicious injuries.
New DNA evidence led Italian prosecutors to reopen the investigation into the 2007 murder of 26-year-old Italian woman.
The nurse is accused of injecting his victims with overdoses of painkillers or sedatives, sometimes in combination with other medications.
In his first comments on the violence, interim President Ahmed al-Sharaa said government forces would pursue "remnants" of the ousted Bashar al Assad government and bring them to trial.
The technology in question was used to solve a murder trial last week.
The investigations reopened after a witness came forward to the antisemitism commissioner for the Bavarian judiciary.
Joseph Czuba, 73, faces life in prison when he is sentenced in May for a murder that prosecutors said was sparked by anti-Muslim hatred.
Rushdie, 77, was stabbed with a knife multiple times in the head, neck, torso and left hand, blinding his right eye and damaging his liver and intestines.