Phrased more generally, you raise the following question: can the rotation of one celestial body affect the state of rotation of another celestial body?
To discuss that, let me make a comparison that will allow me to refer to tabletop demonstrations. The comparison: Gravity and the electrostatic force. Both are spherically symmetric inverse square laws.
We can readily look up tabletop sized demonstrations of electrostatic force. For example when two spheres are electrostatically charged then there is a measurable electrostatic force between the spheres.
Let's take such a setup, and add the following element: all else being kept the same, give one sphere a rotation. Will that rotation affect the other sphere?
I assume you will agree with me that introducing a rotation of one sphere will not have any effect on the other sphere. The reason for that: when a charged sphere rotates around its axis the overall distribution of charge remains the same. Therefore the electrostatic field around the charged sphere does not change in any way.
(It may be that you disagree, and that in fact you expect rotation of one charged sphere to induce rotation of a nearby charged sphere. If that is what you believe I recommend that you go ahead and construct such a setup. It can be done as a tabletop setup; cost can be kept down. If you can demonstrate such an effect heads will turn.)
Back to celestical mechanics:
Astronomical observations inform us that the Sun turns around its own axis. Solar observatories track Sun spots, the Sun spots are seen to revolve; from that observation the Sun's rotation rate is inferred.
Note that there is no way to infer the Sun's rotation from any observation of other bodies. That's because the Sun's rotation affects only the Sun itself, the Sun's rotation does not affect other celestical bodies. Only the overall gravity from the Sun is affecting the other celestial bodies of the solar system.