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if not, why?
What you're seeing is precession, and the turning moment is driven by a torque trying to tilt the wheel, in the case of holding it in one hand driven by gravity.
When you hold one end of the axle, your grip is effectively a free pivot, so it doesn't react the torque generated by the mass being displaced from the fulcrum, so precession takes up the slack. On a Lefty, the joint between the axle and the leg is rigid, so it reacts the gravity driven torque and precession doesn't happen.
gbj_tester-
So if I held it solid and horizontal it wouldn't turn one way. If gravity is pulling down on the wheel and thus causing the precession, would a loaded bike on the ground not produce a force in the opposite direction through the axle upwards and make the wheel turn left? or is the whole thing cancelled by the fact that the wheel is held solid through the axle?
PhilDAS
If you hold a wheel with your hands on the axle and spin it then let go with one hand, the wheel stays upright but also turns the away from the one side holding it.
Therefore, do lefty forks pull slightly to the right? And if not, why?