Practically speaking the president can do anything he wants as long as he is not challenged by congress. The constitution does not explicitly give such authority to issue executive orders. But there is quite a bit of precedent in the use of Presidential executive orders as long as the President is doing so in the execution of the law. When the President issues an executive order that contradicts a law or the Constitution, that becomes a problem. Or in the case of the current President, who seems to think he can simply write the law himself and bypass the legislative branch, it is a real problem. He did this with his own healthcare legislation. And when he has given directives to agencies not to enforce specific laws such as DOMA and areas of our immigration laws, he is directly violating his oath of office. And he is not the first President to do so. Maybe just more blatant than many others in his disregard for the checks and balances. One major difference is that in most past cases of a President refusing to enforce a law, it was based on a belief that a law was unconstitutional. Whereas the President more often uses language like "fairness" rather than cite the lack of constitutionality.
Regardless of whether or not one agrees with the provisions of a piece of legislation from a policy perspective, it is still the job of the executive branch to enforce the law. Not pick and choose which laws they will enforce and simply arbitrarily change them because they want to. If you disagree with a law, then you propose legislation and move it through the legislative branch. If you can't...then too bad. That is why we have a sharing in the balance of power. Ultimately, the President in the real world can do anything he wants as long as he is not challenged by the other branches of gov't. And he may be openly critiqued by his political opponents. But they will never do anything to remove him from office, even if they have the majority control after a mid term election. The political willpower isn't there for an ugly fight. The President knows that, and he knows the majority of the media are on his side. So nothing will change.