Which determines the path that an object in projectile motion follows

The answer is Gravity only. 

A projectile is defined as an object in which the only force acting upon it is gravity. If any other forces are involved the object will no longer be considered as a projectile.

When a projectile is launched in the air, the horizontal velocity remains constant, but vertical velocity is influenced by gravity which pulls until it hits the ground. So the path of the projectile is determined by GRAVITY ONLY.