The woman, in her 20s, who was being treated at the St Richard's Hospital in West Sussex, complained to the staff against a worker.
Police said they received a report from the hospital on January 10 that the woman had made an allegation to hospital staff of rape by the worker, who is not a doctor or a nurse, at the hospital overnight.
Detective Chief Inspector Dave Wardley-Wilkins from the West Sussex CID said: "We are in the early stages of a full investigation. Officers are continuing their enquiries and are receiving full cooperation from Western Sussex Hospitals NHS Trust".
"The victim is receiving help and support from specially trained police officers."
The suspect from Chichester was arrested at around 9 pm on Thursday evening and has been suspended from work.
He remains in custody where he is being interviewed, police said.