In researching this question, I found this page from the Government Publishing Office, which states in 4 U.S.C. §10:
Any rule or custom pertaining to the display of the flag of the United States of America, set forth herein, may be altered, modified, or repealed, or additional rules with respect thereto may be prescribed, by the Commander in Chief of the Armed Forces of the United States, whenever he deems it to be appropriate or desirable; and any such alteration or additional rule shall be set forth in a proclamation.
§1 describes the design of the flag, which I interpret as a rule or custom pertaining to the display of the flag.
I'm not a lawyer though. Can POTUS change the design of the US flag without an act of Congress?