In 1983 he graduated from the Pontifical Catholic University of Argentina with a BA in Political Science, and in 1985 Grossi joined the Argentine foreign service.

[3][6] In September 2015, the Argentine government announced the nomination of Grossi as a candidate for Director General of the IAEA, with support from other countries in Latin America and the Caribbean.

In 2016, however, the government of Mauricio Macri withdrew its support to promote Susana Malcorra's candidacy as UN Secretary General.

[9] In November 2017, after the disappearance of ARA San Juan, Grossi had the idea of reviewing the records of the hydro-acoustic stations of the Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO) as an alternative to obtain clues about what happened with the submarine.

[10][11] The remains of the ill-fated ship were found a year later, about twenty kilometers from the estimated position based on the cited records.

[2] In August 2022, Grossi led a team of IAEA inspectors to the Zaporizhzhia Nuclear Power Plant in Russian-occupied southern Ukraine.