Cali was taken to hospital where he was pronounced dead. Gotti took over the crime organization; he died of cancer, in prison, in 2002.
The Sicily native, who is married to John Gambino's niece, served on the family's ruling panel for several years before being promoted to acting boss in 2015, replacing the previous leader Domenico Cefalu, also known as "Greaseball".
According to the New York Daily News, a family member of Cali's called 911, telling a dispatcher that the gunman's getaway vehicle ran him over.
United States media reported the 53-year-old´s death marked the first murder of a NY mob boss in 34 years, since the death of Paul Castellano - another Gambino family boss - by order of John Gotti.
The Mafia's notoriety and influence have waned in New York City since then, but there have been flare-ups.
Police sources described Cali's murder as "disrespectful" because it happened right outside his family home.
After exiting a limousine outside the restaurant, Castellano and his underboss, Thomas Bilotti, were approached by three men in trench coats who shot them each about six times, The New York Times reported at the time.More news: Giants won’t regret trading for Jabrill Peppers
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It's unclear whether Cali's murder was carried out by a rogue actor or approved by other members of the Italian American mafia in New York City.
"It takes a certain type of evil to murder a friend in their own home, and then console the grieving relatives", said Federal Bureau of Investigation assistant director in charge William Sweeney.
Cali kept a much lower profile than Gotti and was killed in far less spectacular fashion than Castellano.
In 1985, Gambino "capo" John Gotti orchestrated the murder of Castellano to become the most publicized boss of the family. An Italian mobster said Cali was "everything over there" in NY and referred to him as "a friend of ours".
In addition, Probation requested that Cali be forbidden from contacting or associating with those affiliated with any organized crime groups, including the Camorra, 'Ndrangheta, and Sacra Corona Unita factions of the Sicilian Mafia.
Last week, Carmine Persico, the 85-year-old former boss of the Colombo organisation, died after serving 33 years of a 139-year prison sentence. At a 2008 bail, a prosecutor said Cali was seen "as a man of influence and power by organised crime members in Italy".