In a linearly birefringent medium, the splitting of the beams can occur in two ways. It can occur at the surface of the medium. The angle of refraction is then determined by Schells Law. However, it can also occur inside the medium.
A linearly birefringent medium always has an optical axis. Inside the linearly birefringent medium, the electric field of the light can be at a nonzero and nonperpendicular angle to the optical axis. Then, refraction can bend the light beam.
For one polarization of light inside a linearly birefringent medium, the path of the light beam will curve. This is called the extraordinary ray, because light traveling on a curve is extraordinary. The other beam polarization travels on a straight path. This is called the ordinary ray.
I think that you are referring to the behavior of light inside the linearly birefringent medium. The splitting of images in "Iceland spar" is mostly from the internal behavior of the light beams, not the surface refraction. The splitting at the surface occurs but is much smaller in a thick piece of calcite.
The behavior of the extraordinary beam is contingent on the optical axis. If there is no optical axis, both beams act "ordinary".
Refraction can occur at the surface of an optically active substance. This is analogous to the behavior of light at the surface of an optically birefringent surface.
Consider the interface between a medium without optical activity (like air) and a medium with optical activity (levorotary sucrose solution). Refraction can occur at the interface. Thus, unpolarized light at the surface can split into two beams.
Refraction inside an optically active substance is different from the linear birefringent case. The optically active medium usually does not have an optical axis. Thus, light inside the optically active substance travels in straight lines. Refraction does not occur inside the optically active substance, any more than it occurs in a completely isotropic substance. Every ray is an ordinary ray.
Please note that I am ignoring substances that have both linear birefringence and optical activity. An example of this type of substance is crystal quartz. The mixture of the two is rather complicated.
Only at the surface of the optically active (i.e., chiral) substance. Not inside. Light in an optically active substance always travels in a straight line unless it also has linear birefringence.
An extraordinary beam can only occur in a linearly birefringent substance. Optical activity can not cause an extraordinary beam.
I once did a project involving the theory of reflection from an optically active substance. I would say that the reflection from the surface of an optically active substance can be quite "extraordinary". For example, the reflectance coefficient of light normal to an optically active substance does not vary with polarization!
However, that reflection is a different story. The important point here is that optical activity is not associated with an "extraordinary beam". In this way, optical activity is different from linear birefringence.
Just to clarify the terminology: "Optical activity" should be called "circular birefringence". The better known "birefringence" should actually be referred to a "linear birefringence".
Optical activity is caused by an excess of one type of chiral molecule. Achiral molecules and racemic mixtures make up "optically inactive" substances.