The Care Quality Commission (CQC) found patients in the A&E department were waiting in corridors.
It also said they were waiting longer than the standard 18 weeks for surgical specialist treatment.
The RUH said it was disappointed with the overall rating but delighted with its outstanding rating on patient care.
End of life care was also rated as outstanding, which the CQC said was "rare" for a hospital to receive.
But in emergency care, more consistent staffing needed to be in place and patients needed to receive prompt and regular observations.
Patient flow from admission in A&E to discharge, with a social care package in place once they returned home, also needed addressing.
"If we can resolve that and get the flow of patients moving easier that will improve things still further," added inspector Mary Cridge.
In a statement the hospital said it was working with their health and social care partners such as local Clinical Commissioning Groups to improve matters.
The Royal United Hospitals Bath NHS Foundation Trust also runs the Royal National Hospital for Rheumatic Diseases, which was rated overall as requires improvement while its community maternity services were rated as good.
Bath's Royal United Hospital "requires improvement", particularly in the area of patient waiting times, a health watchdog said.