That is the insignia for Pharmacist's Mate 2nd Class, a corpsman. The red cross in the middle distinguishes it from a Petty Officer 2nd Class.
Pharmacist’s mates are petty officers who, under the direction of medical officers, administer medical assistance, treatment, and services to naval personnel. They may serve in the sick bay or dispensary of a ship or shore station, at a naval hospital, or on a hospital ship. When assigned to a marine landing party, they render first aid to the injured on a battle field, or at a dressing station. Many pharmacist’s mates in addition to skill, training and experience acquired in the performance of general hospital duties, take specialized training and through practical experience become competent medical technicians in such fields as X-ray, clinical laboratory, pharmacy, epidemiology and sanitation, fever-therapy, etc. Those pharmacist’s mates who are employed as technicians are required to attend regular classes in general medicine to maintain skills and knowledge previously acquired.
United States Navy Rating Description for the Pharmacist Mate, Second Class, NAVPERS 15386, Navy Department Bureau of Naval Personnel, 1945
You can see an example here of Pharmacist's Mate Second Class George E. Wahlen, USN receiving the Medal Of Honor from President Truman.
Women in WWII could be Pharmacist's Mates via the WAVES. The woman standing in this picture is a Pharmacist's Mate 3rd Class.