I have seen a lot of commentary arguing one way or another regarding this issue. My personal conclusion at this time is "firmly grounded" would be too strong of a statement, in my opinion.
Incorrect. It's a question that only the judiciary is empowered to authoritatively resolve [0]. Were questions of Constitutionality (and whether or not an executive act is both grounded in statute and not prohibited by another statute is, given the Constitutional roles ofmthe legislative and executive branch, ultimately one of the Constitutionality of the act) not one on which the President (and various other officials) were qualified by virtue of their office alone to decide and act on prior to judicial resolution, oaths to the Constitution and/or laws such as those taken by the President and other public officials would be meaningless.
[0] Except that Congress may also do so, but only in the context of impeachment proceedings based on the purported illegality and without binding effect outside the specific proceeding in which it makes the determination. But its decision in such a case is final and unreviewable.
And that is not the case, here. The Executive Order in question is firmly grounded in section 212(f) of the Immigration and Nationality Act.
Get your facts straight.