Bodewitz was born in Gramsbergen on 13 October 1939. Between 1952 and 1958 he attended the gymnasium in Coevorden. He obtained a degree in classical languages at Utrecht University in 1963 and subsequently a further degree in Indo-Iranian studies in 1966.
Bodewitz started working at Utrecht University as an assistant professor of Sanskrit in 1966. He did so for two years and then became an associate professor at Leiden University, which he remained until 1976. In 1973 Bodewitz earned his PhD under Jan Gonda at Utrecht University. In 1976 he was appointed as full professor of Sanskrit and Indo-European linguistics and he succeeded Gonda in that position. In 1988 his teaching assignment was changed to Indian and Iranian languages and cultures, especially Sanskrit. In 1992 his chair was scrapped in a reorganization in which the departments of Sanskrit of Utrecht and Leiden merged. Bodewitz was able to assume the professorship of Sanskrit at Leiden University the same year.
He took up early emeritus status in 2002 due to the introduction of the Bachelor/Master system in the Netherlands. When he left he was the last full professor of Sanskrit in the Netherlands.