DACA was a temporary executive order. Any president has the power to terminate it. Full stop.
So simple even a liberal can understand it.
I'm not arguing that, in this case, the DHS wasn't allowed to do what it did - i.e., implement a policy which, in effect, reversed the DACA policy which was put in place a few years ago.
That said, the legal issue isn't as simple as you are suggesting it is. Regardless of whether the new policy is in itself allowed under our Constitution and other relevant laws, the Administrative Procedure Act imposes requirements - both procedural and substantive - on agency rule-making. If those requirements aren't met, then the rule (or policy) isn't valid. It is on such grounds that the DACA's recision is being challenged.
It may be the case that notice-and-comment procedures were required to be followed in implementing this rule. (Indeed, Judge Hanen issued a preliminary injunction against DAPA and the DACA-expansion on that basis. He blocked those policies because he found, among other things, that it was likely that the DHS didn't jump through the procedural hoops that the APA requires it to jump through in implementing such a policy - not because he found that it was likely that those policies violated the Constitution.)
It's also possible that the DHS' stated reason for rescinding DACA - essentially, that it was itself illegal - was mistaken. If that's the case, then that action may fail other requirements of the APA. It may, in not being based on a legitimate reason, be arbitrary and capricious and as such in violation of the APA.
The point is, it's not enough that a given executive branch action doesn't violate the Constitution or other laws. We have enacted laws - i.e., the Administrative Procedure Act - which further require that, for certain kinds of actions, certain procedures be followed in taking those actions and that certain kinds of reasons be the motivation for those actions. We can argue whether such requirements make sense, but the reality is that they are in place.
I'd also note that DACA wasn't an executive order. I realize that many refer to it as such and that the distinction isn't, depending on the context, particularly significant. The important thing here is that it represents agency rule-making, which possibly subjects it to requirements of the APA and to judicial review, just as its recision does.
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While there is a degree of discretion relating to enforcement of the immigration laws that is given to the Executive (or he has inherently), what Obama did with his EO on DACA was basically re-write the law that the Congress had enacted. That is going way, way too far, and that's why everyone had a big problem with it, including the courts. His selective enforcement and discretion argument was complete BS and a pathetic way to justify to do what he really wanted to do, which was provide amnesty to people here illegally.
I appreciate that many have reached that conclusion - i.e., that DACA didn't represent a proper exercise of executive branch discretion. But based on what the laws actually say, on what long-standing (and accepted) regulations promulgated pursuant to those laws actually say, and on court decisions relating to the extent of executive branch discretion in this area, I think there are reasonable arguments that certain aspects of DACA are properly within the discretion given to the executive branch. I think other aspects of DACA are certainly within that discretion.
We can get into that consideration if you'd like. But before doing so I'd be curious as to which aspects of DACA you think are rightfully within the executive branch's discretion and which aren't. Or do you think that none of them are?
I'd also ask what you mean when you say suggest that courts had a big problem with DACA. We're talking about the original DACA, not DAPA or the DACA expansion. The original DACA was never successfully challenged in court, as far as I'm aware, with most challenges failing on standing grounds. (Even DAPA and the DACA expansion were only blocked based on findings that they likely violated the APA, rather than on findings that they were per se likely unconstitutional.)