Police say 19-year-old Michael Magnan walked up to the cab, demanded money and pulled out a gun in the Flatlands section on Saturday.
Investigators say the driver punched Magnan in the face, prompting him to open fire.
The victim, who was shot as he sat in the back seat with his sister, is now fighting for his life at Kings County Hospital. Nikita Grebelsky, 23, is listed in critical condition.
Magnan is charged with attempted murder.
Police are investigating whether Magnan may have been involved in two other incidents on Saturday, including a shooting that injured four people attending a family gathering and an altercation at a nightclub.
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