Can the nucleus be ionized? Traditionally, the word ionization is conceptualized as the removal of electrons from an atom to leave a net charge on that atom. This is generally achieved through the bombardment of high energy electrons or photons with enough energy to remove an electron from the discrete energy orbitals the electrons form around the nucleus.
The nucleus also contains discrete energy orbitals, similar to electrons. However, the radius of the orbitals are much smaller than that of electrons. Furthermore, it requires much high energy to transition between the energy levels.
Would it be considered ionization to remove a proton, or neutron, from the nucleus by providing enough energy to free a nucleon from the nucleus? Or is there a different name for the interaction?
 A: In the context of nuclear physics this is called photodisintegration.  In such a process, a high-energy gamma ray hits a nucleus and causes a nucleon (or possibly an alpha particle) to be ejected.  For example, you can have the process
$$
{}^2_1 \text{H} + \gamma \to {}^1_1 \text{H} + n,
$$
which can occur for a gamma ray with energy above 2.22 MeV.
A: Protons and neutron carry a baryon number, and protons carry charge.The nuclei are identified by their charge (i.e. number of protons) in the table of nuclei. where the name and number of protons are unique to each nucleon and is called atomic number. The isotopes of each atomic number , depend on the number of neutrons, for  each atomic number. For an ordinary atom, the sum of the atomic number Z and the neutron number N gives the atom's atomic mass number A.
You ask:

Would it be considered ionization to remove a proton, or neutron, from the nucleus by providing enough energy to free a nucleon from the nucleus? Or is there a different name for the interaction?

If you remove a proton by some interaction, what is left will be a nucleus with Z-1 atomic number in the table. If you remove a neutron by some interaction, you will have an isotope of the same name with an atomic mass smaller by the mass of one  neutron.
