No, his soul was already unstable enough to break at six pieces.
The Dark Lord only had his soul split into six pieces (five Horcruxes and the ‘master’ piece of soul) when he attempted to kill the Potters and a piece of his soul broke off when he was ripped from his body by the rebounding Killing Curse. Nagini wasn’t made a Horcrux until years after, when the Dark Lord used Bertha Jorkins’s death to turn her into one. Though it was in only six pieces, not seven, at that point his soul was so unstable that a piece broke off when the Killing Curse hit him.
“He had rendered his soul so unstable that it broke apart when he committed those acts of unspeakable evil, the murder of your parents, the attempted killing of a child. But what escaped from that room was even less than he knew. He left more than his body behind. He left part of himself latched to you, the would-be victim who had survived.”
- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)
The Dark Lord didn’t have all six of his intended Horcruxes made when he went to kill the Potters. He had only made five Horcruxes by then, meaning his soul was only in six pieces. Dumbledore suspected he hadn’t made all and planned to make one with Harry’s death.
“However, if my calculations are correct, Voldemort was still at least one Horcrux short of his goal of six when he entered your parents’ house with the intention of killing you.”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
Further, Dumbledore also had suspected that after the Dark Lord used her to kill an old Muggle man, the Dark Lord then decided to make Nagini his last Horcrux, again meaning she was made into a Horcrux (so the Dark Lord split his soul to make her) only after the Potters were killed.
“After an interval of some years, however, he used Nagini to kill an old Muggle man, and it might then have occurred to him to turn her into his last Horcrux.”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
This is confirmed in an interview with J.K. Rowling, where she says the Dark Lord turned Nagini into a Horcrux using Bertha Jorkins’s death - which happened years after his killing the Potters.
Lady Bella: Whose murders did Voldemort use to create each of the Horcruxes?
J.K. Rowling: The diary – Moaning Myrtle. The cup – Hepzibah Smith, the previous owner. The locket – a Muggle tramp. Nagini – Bertha Jorkins (Voldemort could use a wand once he regained a rudimentary body, as long as the victim was subdued). The diadem – an Albanian peasant. The ring – Tom Riddle Sr.
- Bloomsbury Live Chat (July 30, 2007)
The Dark Lord hadn’t finished creating his Horcruxes when he went to Godric’s Hollow to kill the Potters - he’d already had five of them but he needed one more to make six, and therefore have his soul be in seven parts in all. However, before he ever got to that point, when his soul was still just in six pieces, it was already made so unstable it broke off when the Killing Curse hit him. Not having tried to kill Harry would have helped him because he wouldn’t have been ripped from his body, and he wouldn’t have a piece of soul in his enemy, but it would’ve actually given him a seven-part soul.