That law does not address those already here, and that is where "due process" comes in. They are here legally and have an expectation of free travel just as do you and I. We even offer due process to ILLEGAL aliens present here within our borders. This portion of the order is not only quite likely unconstitutional, it makes no sense. If they meant us harm, they would simply not leave the country.
Those already issued a visa or holding a green card and currently overseas are a question that has not been addressed by the courts. My thought is one of double jeopardy, but the President has wide ranging powers here. He could prevail, even though the inclusion of green card holders is one of questionable use, since they have already undergone fairly extensive checks.
As for blocking new immigration, not only does the law you cite give him that authority, but tons of case law and previous precedents do also. He wins this part in a landslide.
I do believe he will, or should, choose two avenues. I believe that if he reads the court order carefully, he will craft a more specific EO to comply with the letter of the decision. This will give him what he really needs, the ability to pause immigration to set up new standards for background checks and refugee status. Of course, he's Trump and his ego may keep him from doing this and moving on. The second thing I believe he will do is continue his appeal in order to gain clarification from SCOTUS.
