Yes, President Trump can (and arguably is required to) end the unconstitutional issuance of citizenship to babies born on U.S. soil to aliens here illegally.
On October 30, Axios broke a story titled, “Trump targeting birthright citizenship with executive order.” This story touched off frantic debates, propagation of ignorance and bias, and the requisite social media diarrhea most topics do when raised by President Trump.
We must understand that the Constitution is the Supreme Law of the Land, not misapplications of it or historical opinions of the Supreme Court, nor laws passed by Congress and signed by presidents misapplying it. It, the Constitution, is the Supreme Law. We must also understand that a President is required by that same Constitution to ensure it and laws passed in pursuance of it, are enforced.
The Supremacy Clause  says, “…which shall be made in Pursuance thereof….” To describe valid laws that flow from the Constitution. The Supreme Court itself opined that “a law repugnant to the Constitution is void, and that courts, as well as other departments, are bound by that instrument.” [2, 3] This means all governmental departments have a duty to follow the Constitution and laws that align with it, and ignore others. The Constitution may be understood by the average voter  in combination with awareness of, and an unbiased analysis of, related documentation.
The President, and all governmental officers, take an oath of office under Article 6 of the Constitution to uphold it.  Contemporary academia and news media wrongly teach that only the Supreme Court upholds laws. Most are not aware that the Constitution authorizes Congress to prevent the Supreme Court from hearing certain cases , it specifically charges the President with ensuring laws are properly carried out. 
The 14th Amendment was drafted and ratified with the understanding that it was to provide citizenship to persons born on U.S. soil to parents present legally. The Wong Kim Ark SCOTUS case that anchor baby advocates cite clearly shows the subject alien’s parents were legally present at his birth, subject to the jurisdiction thereof,” silent on children of parents here illegally. 
As the 14th Amendment did not delegate to Congress the authority to award citizenship-by-birth to persons born of aliens present illegally, any and all laws attempting to enact such are null and void as if having never existed.
UPDATE: Further proof that Illegal Aliens are not entitled to Birthright Citizenship for their children: Illegal Alien Behavior Supports Legal Argument Against Birthright Citizenship.
1. “This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land.”
— United States Constitution, Article VI, Clause 2
2. “…a law repugnant to the Constitution is void, and that courts, as well as other departments, are bound by that instrument.”
— Marbury v. Madison 5 U.S. 137, 180 (1803)
3. “An unconstitutional act is not a law; it confers no rights; it imposes no duties; it affords no; it creates no office; it is, in legal contemplation, as inoperative as though it had never been passed.”
— Norton v. Shelby County, 118 U.S. 425, Decided May 10, 1886
4. “In interpreting this text, we are guided by the principle that “The Constitution was written to be understood by the voters; its words and phrases were used in their normal and ordinary as distinguished from technical meaning” (Emphasis Added) United States v. Sprague, 282 U. S. 716, 731 (1931)”
— Justice Antonin Scalia writing in District of Columbia v. Heller, June 26, 2008
5. ″The Senators and Representatives before mentioned [federal] , and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution.”
— U.S. Constitution, Article VI, Clause 3
6. — U.S. Constitution, Article III, Section 2, Clause 2, “…The Court’s appellate jurisdiction is given “with such exceptions, and under such regulations as the Congress shall make.” Affirmed via unanimous opinion of the Supreme Court in Ex parte McCardle, 74 U.S. (7 Wall.) 506 (1869)
7. “…he shall take care that the laws be faithfully executed….”
— U.S. Constitution, Article II, Section 3, Clause 5
8. “Birthright Citizenship Is Not Actually in the Constitution” by John Eastman, Henry Salvatori Professor of Law and Community Service, Chapman University’s Fowler School of Law. 22 December 2015 https://www.nytimes.com/roomfordebate/2015/08/24/should-birthright-citizenship-be-abolished/birthright-citizenship-is-not-actually-in-the-constitution