Do the duties of the Secret Service extend to investigating allegations against a White House Medical team member?
It depends on the allegations; but, in this case, no. Secret service agents are "special agents". Special agents have limited authority to conduct investigations and make arrests. Any investigation begun by a special agent stops after it is determined they have no authority to continue. For the Secret Service, the limits are given in 18 U.S. Code § 3056 - Powers, authorities, and duties of United States Secret Service. 18 U.S. Code § 3056 (c)(1)(C) limits the authority to crimes against the United States or felonies.
make arrests without warrant for any offense against the United States committed in their presence, or for any felony cognizable under the laws of the United States if they have reasonable grounds to believe that the person to be arrested has committed or is committing such felony;
Having determined no felony was committed, the special agent then refers the matter to the agency with authority.
The White House Medical Unit , of which RDML Ronny Jackson, MD, was the director, is a military unit. As such, any allegations of misconduct would be investigated by a unit of the Department of Defense under the Uniform Code of Military Justice. Based on the given allegations against Jackson, as a Navy admiral, that authority would have been NCIS. It is not clear that NCIS was called at the time.
"The Senate Armed Services Committee asked the DoD Office of Inspector General (OIG) to investigate the allegations against RDML Jackson on April 26, 2018."  On March 3, 2021, the Department of Defense Inspector General released a report concerning the misconduct allegations.
1 "The White House Medical Unit (WHMU) is a unit of the White House Military Office and is responsible for the medical needs of White House staff and visitors. The unit also provides medical care to the President, the Vice President, their families, and international dignitaries visiting the White House."
2 REPORT OF INVESTIGATION: REAR ADMIRAL (LOWER HALF) RONNY LYNN JACKSON, M.D. U.S. NAVY, RETIRED