This statement is true.
The President of the United States has the authority to strike terrorist organizations that threaten the U.S. because the official role holds the duty to defend liberty, justice, and all that is the United States of America, per the Constitution.
Article II, Section 2 of the Constitution states,
"The President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States; he may require the Opinion, in writing, of the principal Officer in each of the executive Departments, upon any Subject relating to the Duties of their respective Offices, and he shall have Power to Grant Reprieves and Pardons for Offences against the United States"
This specific section of the Constitution authorizes the President of the United States to take action in any way necessary to deter and disrupt any potential acts of terrorism against the United States of America.
The attacks that occurred on 9/11 and devastated the U.S. have furthermore upheld this notion.
"... after the attacks on September 11, 2001, for it is in that time that our government began to articulate the legal framework that we still rely on today. As many of you know, it was only days after the 9/11 attacks that Congress passed, and the president signed, an authorization for the use of military force, or AUMF, authorizing the president to take action to protect the United States against those who had attacked us."
This authorization still holds true to this day.
" The President has the constitutional power not only to retaliate against any person, organization, or state suspected of involvement in terrorist attacks on the United States, but also against foreign states suspected of harboring or supporting such organizations."
This was stated in the Deputy Assistant Attorney General, John C. Yoo's, The President’s Constitutional Authority to Conduct Military Operations Against Terrorists and Nations Supporting Them. This is coming straight from the Department of Justice.
Links:
https://www.justice.gov/file/146041-0/dl?inline=#:~:text=Military%20Operations%20Against%20Terrorists%20and,power%20to%20use%20military%20force.
https://www.senate.gov/about/origins-foundations/senate-and-constitution/constitution.htm
https://ogc.osd.mil/Portals/99/Law%20of%20War/Practice%20Documents/GC%20Preston%20-%20Legal%20Framework%20for%20the%20US%20Use%20of%20Military%20Force%20Since%209-11%20-%20109AmSocyIntlLProc331.pdf?ver=dDHD5jVVldwBBUlzSskTMQ%3D%3D#:~:text=Return%20to%20the%20first%20days,Laden%20and%20his%20terrorist%20organization.
