Electron spin and energy levels of hydrogen atom

If we only account for the coulomb interaction which is the strongest one relevant to the hydrogen atom then the Hamiltonian won’t have any spin dependence. But that’s only an approximation, a very good one nonetheless.

However, if we look carefully enough we see that you are right. Energy does in fact depend on spin. For further details, look at spin-orbit interaction, a relativistic effect. But this is small compared to the coulomb interaction energy. To be precise, $sim 10^5$ times smaller. This extra energy due to spin-orbit causes the spectral lines to spilt. But only when looked at with enough resolution. Hence it is also called as the fine structure.

You can look at this lecture notes from MIT-OCW 8.06 or this hyperphysics page on the fine structure for further details.