  We propose a covariant ghost-free unimodular $F(R)$ gravity theory, which
contains a three-form field and study its structure using the analogy of the
proposed theory with a quantum system which describes a charged particle in
uniform magnetic field. Newton's law in non-covariant unimodular $F(R)$ gravity
as well as in unimodular Einstein gravity is derived and it is shown to be just
the same as in General Relativity. The derivation of Newton's law in covariant
unimodular $F(R)$ gravity shows that it is modified precisely in the same way
as in the ordinary $F(R)$ theory. We also demonstrate that the cosmology of a
Friedmann-Robertson-Walker background, is equivalent in the non-covariant and
covariant formulations of unimodular $F(R)$ theory.
