WWII Supreme Court Cases
During WWII, the Supreme Court dealt with three main issues in their cases on Executive Order 9066: curfews, exclusions, and internment of persons with Japanese ancestry.
In the lesser-known Kiyoshi Hirabayashi v. United States case, the court had to determine whether imposing a curfew on those of Japanese descent was valid. The main constitutional issue ended up being the effectiveness of the Fifth Amendment in providing for due process of law in times of war. In the end, the court decided that a curfew was indeed justified.
Following this, the majority of the court found in Korematsu v. United States, that the exclusion of those with Japanese ancestry from certain military zones (in this case, the West coast) was also constitutional. In their decision, they heavily relied upon the Hirabayashi case.
The question of internment of people with Japanese ancestry was not actually decided constitutionally. Another case, a habeas corpus case, Ex parte Endo might have resolved it, but the Court concluded this case in favor of Endo without going to the constitutional level. A day before this decision was announced, the government suspended Executive Order 9066, leaving the question unresolved.
In these cases, Justice Frank Murphy's opinions stand out. In Hirabayashi, he sternly warned of the racism involved, but felt that the curfew was indeed justified. However, in Korematsu, he evidently felt that the exclusion was without basis and dissented, calling the majority opinion "legalization of racism."
Current Law
In theory, these cases still stand. This is because they haven't been explicitly overruled, and the changes to the constitution since Hirabayashi aren't relevant: Amendments XXII and XXV have to do with procedural issues with the office of the President, amendments XXIII, XIV, and XXVI are related to voting rights, and XXVII addresses Congressional salaries. The constitution for the purpose of a similar order is thus unchanged, and so should theoretically be covered under Korematsu & Hirabayashi.
However, its effective status as precedent is shaky. For one, it has been widely condemned, and courts would likely avoid using Korematsu & Hirabayashi if possible, or attempt to make a distinction from them given a slightly different order. Second, the convictions against Korematsu and Hirabayashi were overturned in the 1980s based on evidence being possibly concealed by the government. This was confirmed by the Department of Justice in 2011. However, this is an error of fact, and as such the ratio decidendi (legal reasoning) used would still technically stand as precedent.
Lengthy Update: The Supreme Court has since declared Korematsu incorrect in Trump v. Hawaii. However in doing so, the court explicitly distinguished the two cases. Thus, many consider the declaration to be obiter dictum, i.e. an aside that is non-precedential. I'll quote the relevant portion so that readers may decide for themselves:
Finally, the dissent invokes Korematsu v. United States, 323 U.S. 214, 65 S.Ct. 193, 89 L.Ed. 194 (1944). Whatever rhetorical advantage the dissent may see in doing so, Korematsu has nothing to do with this case. The forcible relocation of U.S. citizens to concentration camps, solely and explicitly on the basis of race, is objectively unlawful and outside the scope of Presidential authority. But it is wholly inapt to liken that morally repugnant order to a facially neutral policy denying certain foreign nationals the privilege of admission. [...]
The dissent's reference to Korematsu, however, affords this Court the opportunity to make express what is already obvious: Korematsu was gravely wrong the day it was decided, has been overruled in the court of history, and—to be clear—"has no place in law under the Constitution."
Additionally further muddying the issue is that the judgment directly moves from referencing Korematsu to referencing concentration camps. But as noted above, Korematsu dealt with exclusion, not internment in concentration camps as is often popularly believed (though Murphy's & Roberts's dissents dispute the distinction so the Trump v. Hawaii judgment may implicitly be validating this). Korematsu paragraph 20:
It is said that we are dealing here with the case of imprisonment of a citizen in a concentration camp solely because of his ancestry, without evidence or inquiry concerning his loyalty and good disposition towards the United States. Our task would be simple, our duty clear, were this a case involving the imprisonment of a loyal citizen in a concentration camp because of racial prejudice. Regardless of the true nature of the assembly and relocation centers—and we deem it unjustifiable to call them concentration camps with all the ugly connotations that term implies—we are dealing specifically with nothing but an exclusion order.
Future Challenges
If another such order occurred, it would almost certainly reach the Supreme Court. The Court would then have to adhere to, reconcile with, or overrule Korematsu & Hirabayashi. In Hirabayashi, the following key piece of reasoning was given (and quoted in Korematsu):
Whatever views we may entertain regarding the loyalty to this country of the citizens of Japanese ancestry, we cannot reject as unfounded the judgment of the military authorities and of Congress that there were disloyal members of that population, whose number and strength could not be precisely and quickly ascertained. We cannot say that the war-making branches of the Government did not have ground for believing that in a critical hour such persons could not readily be isolated and separately dealt with, and constituted a menace to the national defense and safety, which demanded that prompt and adequate measures be taken to guard against it.
Regardless of personal beliefs, it is hard to argue the logic behind this particular statement (mainly because it is so broad, in my opinion). However, these "prompt and adequate measures" would still have to be weighed against the Fifth Amendment. Additionally, we now have a historical example of what may happen if race-based exclusion were allowed, and this could inform the justices' decision.
A completely different way that such an order could be ruled unconstitutional could be through separation of powers violations. It could be ruled that Congress cannot constitutionally delegate the powers expressed in Executive Order 9066, but this is rare. It was mentioned in Hirabayashi, but the court did not consider this issue, saying that since Congress agreed with the order, this point was moot.
Conclusion
Though the letter of the law suggests that another order like Executive Order 9066 is possible, in practice it would face significant hurdles. Such an order would likely face challenges based on the Fifth Amendment and almost certainly make its way to SCOTUS. Then, it would be up to them to decide.
I'll end with this quote from the late Justice Antonin Scalia:
Well, of course, Korematsu was wrong [...] But you are kidding yourself if you think the same thing will not happen again.
