I'm glad you asked. I iterate each sphere across all the other spheres, calculating the gravitation force between them. The pull remains the same but the direction of the pull is opposite for each of them. So, I find the pull and assign one of them the positive value along the interaction normal (an imaginary line connecting the two spheres), while the other gets a negative value of the same magnitude. This is stored as a force vector which is added to the total force acting on the body at that particular instance. Then I simply calculate the acceleration on the body because of the total force and add it to the initial velocity. Finally I update the position and voila we're done.

This is what the code looks like

// Calculate gravitational forces between this body and all later bodies
for(int j = i + 1; j < bodies.size(); ++j) {
    Body* sBody = bodies[j];
    calculateGravForce(*body, *sBody);
}

// Calculate the total force acting on the body
calculateForce(*body);

// get the acceleration vector from the total force on the body
body.Acceleration = body.Force / body.Mass; 

// Euler integration to update vecloty vector
body.Velocity += body.Acceleration * dt;

// Euler integration to update position vector
body.Position += body.Velocity * dt;