- In one sense, " spiritual" relates to deep feelings and beliefs, in particular religious beliefs. The " spiritual life " of a person is the history of her " inner self" inasmuch as this person cares for this "self / soul " ( in latin, cura sui) , wants to improve herself from a moral standpoint and to participate in a kind of union with values that she considers as above herself ( maybe as transcendent or divine ).
To this extent, metaphysics does not deal properly with the spiritual . But some metaphysical systems aim at providing rational grounds for the " spiritual" attitude as it has been discribed.
Now, from a philosophical standpoint, " spiritual" relates to " intellectual substance" ( substance meaning here : fundamental entity) , that is to a substance ( entity) that is able to operate actions independently from matter , in particular the operations of knowing and of willing ( loving being included in willing in a broad sense).
So the question is : is metaphysics necessarily committed to the existence of spiritual entities ( be they human souls, or angels, or God)?
If you consider the whole history of philosophy the answer should be " no": negating the existence of " souls" or of "God" is a metaphysical thesis (claim) as much as claiming their existence; there is a materialistic or an atheistic metaphysics as well as a spiritualist one.
However one should also say that still historically, mainstream metaphysics ( Plato, Aristotle Descartes, Leibniz, Wolff, even Hegel) is opposed to materialism.
I think a consensus could be reached by saying that metaphysics deals not with spiritual matters properly, but with intellectual
things. " Spiritual " is only one possible understanding of "
Let me distinguish 3 meanings of " intellectual"
(1) What is remote from sense experience, what is abstract, general, universal : every school of metaphysics could admit that metaphysics deals with what is " intellectual" in this sense. It means that metaphysics aims at answering the question : " what is being? " in the most general way. This is metaphysics understood as "ontology".
(2) What provides intelligence with the satisfaction of its intrinsic need, namely understanding. In that sense, metaphysics is the study of causes and principles, and more precisely, the highest causes or the most fundamental ones. One way to answer the question of " ultimate causes" is to say that the first cause of the universe cannot be material, but has to be " spiritual" , more precisely, God. On this understanding metaphysics is called " first philosophy".
(3) What has an intellectual mode of being, what is , by nature, free from matter ( human souls, angels, God). In this third sense, metaphysics can be said to deal with what is " spiritual" and is called " ( natural) theology".
My answer is borrowed from Aquinas, " On the nature and excellence of metaphysics" https://archive.org/details/introductiontome00inthom/page/15/mode/1up