Officers were called to the Tesco store on Altrincham Road in Baguley at about 18:45 BST on Monday, following reports a baby had been hit.
Mother Amy Duckers said it was the first time the family had taken baby Elsie Rose out since she was born.
A 63-year-old arrested on suspicion of assault is currently in custody being questioned by officers.
Ms Duckers said the suspect had apologised and said he thought the baby was a doll.
Ms Duckers was showing her new daughter off to friends and neighbours in the store with her partner Lewis Temple, and her seven-year-old daughter, Libby.
She said a neighbour saw a work colleague who was with her husband, and it is alleged he came over and punched the baby, leaving a "very big fist mark" on her forehead.
The "mortified" parents called security and the baby girl was taken to hospital as a precaution. She has since been discharged.
Ms Duckers praised Tesco staff at the store saying the way they had dealt with the situation was "amazing".
A spokesman for the store said: "We were deeply shocked and saddened to hear about this traumatic incident and our thoughts are with baby Elsie, Amy and her family.
"We're grateful for Amy's kind comments about our colleagues who we commend for their actions and we will be assisting the police with their investigation."
A man was arrested on suspicion of assaulting a five-day-old girl at a supermarket in Manchester, police said.