How to rotate the transform only on the Y?

private Update()
{
// Here the transform player should rotate facing the opposite direction.
// Determine which direction to rotate towards
Vector3 targetDirection = targetToRotateTo.position - transform.position;

// The step size is equal to speed times frame time.
float singleStep = rotationSpeed * Time.deltaTime;

// Rotate the forward vector towards the target direction by one step
Vector3 newDirection = Vector3.RotateTowards(transform.forward, targetDirection, singleStep, 0.0f);

// Draw a ray pointing at our target in
Debug.DrawRay(transform.position, newDirection, Color.red);

// Calculate a rotation a step closer to the target and applies rotation to this object
transform.rotation = Quaternion.LookRotation(newDirection);
}


If your object always remains upright (green vector always points up), then this is as simple as zeroing out the y component of your target direction.

Vector3 targetDirection = targetToRotateTo.position - transform.position;
targetDirection.y = 0f;

if (targetDirection != Vector3.zero) {
float singleStep = rotationSpeed * Time.deltaTime;

transform.rotation = Quaternion.RotateTowards(
transform.rotation,
Quaternion.LookRotation(targetDirection),
singleStep);
}


If your vertical axis might point in an arbitrary direction, then you can use the trick I've explained a few times in past Q&A: