We study the conformable fractional (CF) Dirac system with separated boundary conditions on an arbitrary time scale mathbb{T}. Then we extend some basic spectral properties of the classical Dirac system to the CF case. Eventually, some asymptotic estimates for the eigenfunction of the CF Dirac eigenvalue problem are obtained on mathbb{T} . So, we provide a constructive procedure for the solution of this problem. These results are important steps to consolidate the link between fractional calculus and time scale calculus in spectral theory.