Morabia completed his undergraduate studies at Collège Calvin in Geneva in 1971, majoring in Greek and Latin. After receiving his M.D. from the School of Medicine at the University of Geneva in 1978, Morabia trained in internal medicine at the University Hospital of Geneva and in occupational medicine in Italy. He is board certified in both internal medicine and occupational medicine. In 2009, he was appointed Fellow of the Royal College of Physicians of Edinburgh.
In 1986, Morabia received a grant from the Swiss National Science Foundation to study at the Johns Hopkins School of Hygiene and Public Health, where he obtained M.P.H. and Ph.D. degrees in epidemiology, the first such PhD awarded to a Swiss citizen, and an M.H.S. in biostatistics.
In August 1990, he became chair of the Clinical Epidemiology Unit at the University Hospital of Geneva, the first epidemiology group ever created in a Swiss hospital. Under his leadership, the unit grew into a division, and he was subsequently appointed professor of clinical epidemiology at the University of Geneva.