There are three types of people on the list: 1. US citizens; 2. resident aliens; and 3. foreign nationals who are not resident aliens. The third category is by far the largest. Frank will correct me if I am wrong, but my understanding is that only the first two categories are legally allowed to buy firearms from or through an FFL. And that is all we are really talking about, since private sales in the states that allow them (and all illegal sales for that matter), do not require background checks, and thus the screening system of NCIS would necessarily fail.
So what we are left with is persons legally residing in the US who, of most critical importance, have no disqualifying misdemeanors or any felonies, nor have been accused and are under indictment for a disqualifying offense. As far as the law (and the Constitution) is concerned, these people are, by definition, innocent of any crime (at least until arrested, tried and convicted)., yet this proposed law would bar them from the exercise of a Constitutionally guaranteed right. To deprive persons from the exercise of any right, i.e., to deprive them of life, liberty or property (meaning a second amendment right to keep and bear arms), without due process of law (meaning a conviction) is a violation of the Fifth Amendment and of the Fourteenth Amendment. Any such law would be therefore facially unconstitutional.
To add to the illegality, the government refuses to divulge what it takes to get on the list, and for years refused to allow anyone on the list any remedy to allow them to get off. According to tweets in response to the President's message, 72 TSA employees are on the list, as was, famously, Senator Ted Kennedy. There have been estimates that 40% of the people denied boarding are on the list in error (e.g. children and the aged), have the same (or most egregiously similar) name to someone on the list, have been added to the list based on unfounded accusations or suspicions, or because of their religious affiliation, travel, or business activities abroad, despite no evidence at all of illegal conduct.
To make it simple, if you don't have enough evidence to arrest someone for a criminal act (in legalese, have probable cause to arrest) then you have no authority, moral or legal, to deprive them of their Constitutionally guaranteed rights.