One’s religion is relative to his election, as there is no real separation from one’s beliefs and actions, as the latter are overall determined by and manifest what one really believes. The “religion” of most politicians is that of political pragmaticism, by which getting elected largely determines one’s positions.
In re. to Romney, while a true Christian is to hold the Bible as his ultimate authority, and thus we can be sure one can not take up arms in seeking to defend or expand the faith if acting consistent with the New Testament (i am not speaking of the possibility of protecting innocent life in an immediate threat, or in a police type job), the case of a person who professes faith in a religion whose heads claim authority over the Bible is a different case. Under such Rome took upon much of the form of the Roman Empire and much used its means to unBiblically to achieve it’s ends, while the Mormons definitely also acted unBiblically in it’s Utah war*. Etc. Just as her fantastic fable in the BOM and her doctrines are. And thus a profession to uphold the Constitution as well as be faithful to his faith is problematic (as is a faithful Christian being President, but due to the kind of moral comprise required).
And to a much lesser degree, even the Pilgrims acted rongly in seeking to establish a theocracy, as Roger Williams realized. And changed, as the Bible supports separation of church and state (Mt. 22:17-21; Jn. 18:36; 1Cor. 5:12, 13), which Jesus and the early church exampled. Christian theocracies fail, as the New covenant does not support such.
Of course, it is impossible to completely separate beliefs from an educational or legal system, as they are founded upon such. And presently secularism has much displaced the morality based upon general Christian faith.
*http://www.rickross.com/reference/mormon/mormon320.html
http://www.globalsecurity.org/military/ops/utah.htm