Really he has the legit power to stop deportations?
When resources are limited, you cannot enforce everything to the letter. The agencies involved, report to the president, through his politically appointed Secretary.
It is like a city police department. The Mayor calls the police chief, and says, "look, let's clamp down on drunk driving, and let the jaywalking slide if we have to in order to free up the cops."
So yes, the President has the ability to arrange priorities. Such has long been the case; e.g.. before Obama.
These protesting groups usually march to his tune. That there is conflict, displeasure seems like a good thing, for November elections.