In this study, we analyze a conformable fractional (CF) Sturm-Liouville (SL) equation with boundary conditions on an arbitrary time scale $\mathbb{T}$. Then we extend the basic spectral properties of the classical SL equation to the CF case. Finally, some sufficient conditions are established to guarantee the existence of a solution for this CF-SL problem on $\mathbb{T}$ by using certain fixed point theorems. For explaining these existence theorems, we give an example with appropriate choices.