Grant it I have not read every single response to this post, so I am not sure if someone else has mentioned this, but I think the ship's counselor is the closest thing that you would find on a Federation star ship.
Have to keep in mind that Starfleet is made of up different species with different beliefs (even if they don't believe in anything) and that diversity would have to be respected.
Even in the real world, in U.S. Navy vessels, and throughout the U.S. military there are not just Christian Chaplains, but Muslim Chaplains, Jewish Chaplains, etc. to reflect the religious diversity of service personnel. So one could imagine that in an interstellar environment there would be even GREATER diversity of belief. No counselor could or should represent any singular belief.
As for those who say that spirituality is not dealt with in Star Trek, I suggest looking to Deep Space Nine (my favorite of the whole ST genre) whose whole story arch from Season 1 to the series finale dealt directly with spiritual belief and spiritual conflict - the Bajorans. There were episodes that dealt with creationism vs. evolution, religious extremism, zealotry, etc.