Is it possible to estimate the 23.5° tilt of Earth's axis? Using the value of the precession of Earth's axis in 25,800 years and the value of Earth's radius, mass, moment of inertia etc., would it be possible to calculate/estimate that Earth's axis must be tilted approx. 23.5°?
The question can probably be posed in different ways. E.g. is the precession period related to the tilting angle?
 A: The precession of the earth's tilt is caused primarily by the torque from the moon pulling on the equatorial bulge (with periodic help from the sun).  The pull of each is stronger on the near side bulge. The torque varies throughout a month and year and will be strongest when the earth has its maximum tilt toward the moon and sun. This gets complicated because the moon's orbit is tilted relative to that of the earth.
A: First some general remarks:
In real life astronomy it is the moment of inertia that is known to the least level of precision.
Celestial bodies don't have a uniform density.
If the goal is to calculate the moment of inertia taking the the density as a function of distance-to-the-center into account, then you need those density data.
In real life astronomy: whenever necessary the moment of inertia is inferred from the rate of precession.
Journal article:
Information on internal structure from shape, gravity field and rotation


As to your question specifically: Let's use the name 'C' for any celestial body. The rate of precession arises from the magnitude of the torque that C is subject to. The magnitude of the tilt does affect the magnitude of the torque that C is subject to.
A possible snag: what if there are, for a given rate of precession, two different tilts that will give rise to that rate of precession? Just of the top of my head I can't exclude that possibility.
With that caveat: it is the case that the magnitude of the tilt affects the torque that C is subject to.
