The debate about executive power is a critical one to have, that's been really brewing for probably about 14-16 years or so I would say just off the top of my head, and isn't limited to this particular issue.

However, this particular debate has to be on the merits. If the fourteenth amendment is found to not protect birthright citizenship, then the President actually does have the power to limit it by executive order. Nearly all immigration functions fall under the executive branch, unless Congress chooses to legislate on the issue -- on this particular issue I can't recall a specific law that addresses it and I suspect that given current leanings in the senate at least it would be very difficult to pass such a law.