The victim, aged 32, was hit by the car at about 11:30 BST on Friday in the McDonald's car park at China Garden, on Derby Street, in Bolton.
Police said he and friends were in a confrontation with another group.
A man, 19, has been arrested on suspicion of dangerous driving and both failing to stop and report a collision.
Officers said  four people the victim had been arguing with left in two cars but no-one in the vehicles stopped.
He is in a serious but stable condition in hospital.
Greater Manchester Police is now investigating the circumstances leading up to the crash, including allegations of racist remarks made during the argument.  Officers are also analysing CCTV images.
A spokesman for the force said they "were certainly investigating the possibility" that the car was deliberately driven at the victim.
A man has suffered severe fractures after a car crashed into him following a row in a car park in Greater Manchester.