Another tack.
Why is Ohm's law called a "law"? Because the data show that it is always obeyed. When some function or attribute is called a law in physics, it means that within that mathematical framework it is the axiom needed to be able to model the data mathematically. It means that it has been observed experimentally that the data always obey this law, period. This is more evident in the laws that Maxwell's equations assume as axioms in order to extract from general differential equations , those solutions that obey the laws that were found to be always "obeyed" by the data.
This is where I am getting a little confused -- I understand how a voltage can cause a current, but why does a current necessarily cause a voltage (that still satisfies V=IR)?
In this simple classical case of having to solve algebraic equations, it means that Ohm's law can be used safely when two of the variables are given to find the third. It is a law, which means that there has been no measurement or observation that falsifies it, within the classical framewor.
When one goes to the underlying framework of atoms and molecules and quantum mechanics, one can see the derivations of ohm's law within the more rigorous framework:
A microscopic view suggests that this proportionality comes from the fact that an applied electric field superimposes a small drift velocity on the free electrons in a metal. For ordinary currents, this drift velocity is on the order of millimeters per second in contrast to the speeds of the electrons themselves which are on the order of a million meters per second. Even the electron speeds are themselves small compared to the speed of transmission of an electrical signal down a wire, which is on the order of the speed of light, 300 million meters per second.
there is further analysis in the link