No, the NTSB will carry out the investigation, after (or during) which they will make any Safety Recommendations that they consider appropriate, directed to the airline, manufacturer, regulator, etc.
It's not so black-and-white.
The NTSB is a relatively small agency tasked to handle 1,500+ accidents and incidents each year. It just doesn't have the manpower to investigate each case by itself.
So when an accident or incident is serious but does not involve fatalities, the NTSB will routinely delegate the field investigation work to the FAA. I.e., the FAA will collect and take into custody all the evidence including photographs, any wreckage, CVR/FDR data, crew/witness interviews, etc., then forward them to the NTSB for final analysis and determination of Probable Cause.
Plus in this case, Qatar is a Part 129 carrier under supervision from an FAA International Field Office (IFO Frankfurt, until the end of the month).