The accident happened Friday night on Eighth Avenue in midtown.
Police said 40-year-old Clark Gettinger was driving a 2001 Lexus northbound when he first hit a taxi and then hit the bicyclist from behind.
Deliveryman Ricardo Gonzalez fell off the bike and slid under an SUV.
The 27-year-old Gonzalez was taken to Bellevue Hospital in critical condition.
Witnesses told police that Gettinger never slowed down. He was later pulled over and arrested.
Gettinger pleaded not guilty on Saturday to charges including driving while intoxicated and leaving the scene of an accident.