Different points in earth are at slightly different distances from the sun and hence experience different forces due to gravitation. For a rigid body, we know that if various forces act at various points in it, the resultant motion is as if a net force acts on the c.m. (centre of mass) causing translation and a net torque at the c.m. causing rotation around an axis through the c.m. For the earth-sun system (approximating the earth as a uniform density sphere)

To determine the effect of the gravitational force exerted by the sun on different points of the earth, we model the earth as a uniform density sphere.

The gravitational force on any small mass element of the earth is directed towards the center of the sun. Because the earth is assumed to be a uniform density sphere, it exhibits perfect spherical symmetry.

For a spherically symmetric body, the net gravitational force exerted by an external mass (the sun) acts as if the entire mass of the sphere is concentrated at its center of mass. The line of action of this resultant gravitational force passes directly through the center of mass of the earth.

Recall that torque ($\tau$) is defined as the cross product of the position vector ($r$) from the axis/center of rotation to the point of application of the force, and the force vector ($F$):
$\tau =r\times F$
Since the line of action of the net gravitational force passes directly through the center of mass of the earth, the position vector from the center of mass to the line of action of the force is zero (or parallel to the force). Thus, the net gravitational torque acting about the center of mass of the earth is zero.

Therefore, the correct option is the torque is zero.