Vespa received his degree from the University of Florence in 1751. In 1757, he received a state scholarship to study at the Institute of Obstetrics André Levet in Paris. Upon his return to Florence, he taught obstetrics at the hospital of Santa Maria Nuova. In 1761, he published the Trattato dell'arte Ostetrica. He was appointed lecturer at the University of Pisa in 1769. In 1772, he was made Court Obstetrician, and relieved of his duties at Santa Maria Nuova. In 1790, he followed the newly elected Emperor, Peter Leopold, to Vienna.