What is the name of the hospital where Sherlock Holmes and Dr. Watson first met?

The hospital where Sherlock Holmes and Dr. Watson first met was St. Bartholomew's Hospital.

In A Study in Scarlet, Watson returns to London after serving as an army medical officer in Afghanistan. His old acquaintance Stamford tells him about a peculiar man seeking a roommate, then takes him to a chemical laboratory at St. Bartholomew's Hospital, where Holmes demonstrates his famous blood-detection experiment. Their conversation leads directly to their partnership and eventual residence at 221B Baker Street.

St Bartholomew's, commonly called Barts, is one of London's historic hospitals. Founded in 1123 by Rahere, it remains associated with medical education and practice. Watson had studied there before qualifying as a doctor, which explains why Stamford could introduce Holmes to him in that setting.

A common mix-up is St. Thomas' Hospital, another major London institution. It is not the meeting place in Doyle's original story. The first meeting occurs at Barts, although later adaptations sometimes alter details of the introduction.