What force is most responsible for the path of an object during projectile motion?

The gravitational force.

In fact, the motion of the projectile is composed by two independent motions:

- on the horizontal direction, it is a uniform motion (with constant speed)

- on the vertical direction, it is a uniformly accelerated motion, where the vertical acceleration g is given by the gravity exerted by the Earth on the projectile.

For this reason, the composition of the two motions results in a parabolic trajectory.