There are obvious Constitutional difficulties inherent in having what is, if not the, certainly one of the most visible and respected parts of the US Executive becoming so obviously partial toward a particular religious congregation. It is also an arena which is directly on point as to why the Founding Fathers wanted to raise the Wall of Separation in the first place; you do not want all the serious firepower concentrated in the hands of a particular religious faction!
I would add that giving over all the big guns to an emotive, evangelizing and millenarian style of religion is a particularly dangerous idea; this isn't paranoia, history books contain many examples of states which have been overthrown by militaries in the thrall of such ideologies.
On a smaller level, spare a thought for individuals within the military who aren't of the dominant religion, as was my own situation while I was in.
As a military subject you are usually a captive audience; on board a ship, or in a barracks, there is no easy way to escape the proselytizing. I can't count the number of times on board ship when I was forced to sit through lengthy prayers and even full religious services, broadcast over the ship's loudspeakers into every corner.
Consider also that a military subject, by the nature of the profession, is often forced to live and work in continual close proximity with other members of his or her unit, under extremely stressful and unnatural conditions. Not only success in mission or career, but sometimes life itself is at stake, and dependent upon being able to develop and maintain close and trusting relationships. If a majority of the unit belongs to a particular faith, and is encouraged or simply allowed to push it onto the minority, it is difficult for a civilian to even imagine the overwhelming social pressures which can be brought to bear.
If impartiality of religion is not vigorously enforced as a principle of military life, eventually you will end up with a self-enforcing culture; only Christians will be able to effectively work in the unit, because the existing members won't trust or bond with a non-Christian. There's a reason why Iraqi police and military units aren't typically mixed Shia and Sunni; the analogous issue fails of becoming a problem for US forces by a much smaller margin than you might imagine. Take it from one who's been there.