The Catholic Archbishop for the military diocese, Archbishop Timothy Broglio, released a statement at the request of the military Chief Chaplains on the impact of a repeal of DADT on the Chaplaincy.  He had some strong and unequivocal words on the subject: 

Those with a homosexual orientation can expect respect and treatment worthy of their human dignity…However, unions between individuals of the same gender resembling marriage will not be accepted or blessed by Catholic chaplains. Furthermore no restrictions or limitations on the teaching of Catholic morality can be accepted.  First Amendment rights regarding the free exercise of religion must be respected.

This means that Catholic chaplains must show compassion for persons with a homosexual orientation, but can never condone-even silently-homosexual behavior.  A change might have a negative effect on the role of the chaplain not only in the pulpit, but also in the classroom, in the barracks, and in the office…

This Archdiocese exists to serve those who serve and it assists them by advocating moral behavior.  The military must find ways to promote that behavior and develop strong prohibitions against any immoral activity that would jeopardize morale, good morals, unit cohesion and every other factor that weakens the mission.  So also must a firm effort be made to avoid any injustices that may inadvertently develop because individuals or groups are put in living situations that are an affront to good common sense.

Sacrificing the moral beliefs of individuals or their living conditions to respond to merely political considerations is neither just nor prudent especially for the armed forces at a time of war.

Also noted at the Religion Clause.