Personally, I don’t agree that Islam is the greatest threat facing America. If anything, many Muslims probably have a closer conception of the moral order and certain elements of the common good with the Catholic conception than a good chunk of the US. In my experience, however, most–like most Catholics–are indistinguishable from any other American, for better or worse.
As a general point, at least under Catholic doctrine, religion is something that can be taken into account when regulating immigration, just as it can when regulating religious freedom (see CCC 2109 for this principle in that context).
As the Catechism notes: “Political authorities, for the sake of the common good for which they are responsible, may make the exercise of the right to immigrate subject to various juridical conditions” (CCC 2241). In fact, the whole point of public authority is " to ensure as far as possible the common good of the society." (CCC 1898).
The common good is “the sum total of social conditions which allow people, either as groups or as individuals, to reach their fulfillment more fully and more easily” (CCC 1906) and, among other things includes a society’s “spiritual goods” (CCC 1925). St. John XXIII summed this up as follows:
- In this connection, We would draw the attention of Our own sons to the fact that the common good is something which affects the needs of the whole man, body and soul. That, then, is the sort of good which rulers of States must take suitable measure to ensure. They must respect the hierarchy of values, and aim at achieving the spiritual as well as the material prosperity of their subjects.(42)
- These principles are clearly contained in that passage in Our encyclical Mater et Magistra where We emphasized that the common good "must take account of all those social conditions which favor the full development of human personality.(43)
- Consisting, as he does, of body and immortal soul, man cannot in this mortal life satisfy his needs or attain perfect happiness. Thus, the measures that are taken to implement the common good must not jeopardize his eternal salvation; indeed, they must even help him to obtain it.(44)
Inasmuch as a religion denies faith in Christ or draws souls away from His Body, it does not help one obtain salvation. For example, a country that was blessed to be united in the true faith could legitimately limit the immigration of non-Catholics if there were a sufficient risk that this unity would be substantially broken and souls led astray. This kind of situation, however, is clearly nowhere near the situation in the US and would not override the general principle that the whole earth was given by God for the sustenance of all men (the basis of the right to immigrate). Given the current makeup of society, accepting those who at least honor God as Creator is probably a net gain IMO, all else being equal. Of course, there are a myriad of other factors at play as well in the immigration debates.