1st. This is NOT a question on the merits of an immigration ban. It is a question on procedures.
Suppose the President put out a memo telling Immigration officials to enforce all of the aspects of immigration law and procedures to the nth degree to all potential immigrants from <insert country here>. We all know how sticky laws and bureaucracies can be, and the time it takes to move through such places. So would not this have the same effect that Mr. Trump is seeking, without all the drama of an outright ban ?
I don’t believe that there are current laws banning the entry into the US by foreign visitors of the specific countries that Trump’s executive order in question is attempting to do. So POTUS issuing an order for all agencies to enforce the current immigration laws on the books wouldn’t achieve that objective, but there would be several million illegal immigrants from various countries, primarily Mexico, that have overstayed their tourist visas, rounded up and deported.
So, is there a gray area where the President issue a directive for selectively strict enforcement of existing legislation against certain specified countries?
The main problem with the travel ban is that it has a constitutionally improper motivation–a desire to target Muslims. Any action taken with that motivation is subject to challenge under the Establishment Clause by someone with standing to do so, which includes selectively targeting countries for extreme vetting.