Melia was born in Gorizia, Italy. He was educated at Melbourne University and The Massachusetts Institute of Technology, and held a post-doctoral research position at the University of Chicago before taking an assistant professorship at Northwestern University in 1987. Moving to the University of Arizona as an associate professor in 1991, he became a full professor in 1993. From 1988 to 1995, he was a Presidential Young Investigator (under President Ronald Reagan), and then an Alfred P. Sloan Research Fellow from 1989 to 1992. He became a Fellow of the American Physical Society in 2002. He is also a Professorial Fellow in the School of Physics, Melbourne University.

From 1996 to 2002, he was a Scientific Editor with the Astrophysical Journal, and since then has been an Associate Editor with the Astrophysical Journal Letters. He is also the Chief Editor of the Theoretical Astrophysics series of books at the University of Chicago Press.

In a career that has seen him publish over 230 research papers and several books, Melia has made important contributions in High Energy Astronomy and the physics of supermassive black holes. He is especially known for his work on the galactic center, particularly developing a theoretical understanding of the central supermassive black hole, known as Sagittarius A*.With his students and collaborators, he was the first to propose imaging this object with millimeter-interferometry, which should be feasible within a few years, proving beyond any doubt that it possesses an event horizon, as predicted by Einstein's theory of general relativity. He is also a well-respected and popular publicist of astronomy and science in general, delivering many lectures at public venues, including museums and planetariums. His books have won several awards of distinction, including the designation of Outstanding Academic Books by the American Library Association, and selection as worldwide astronomy books of the year by Astronomy magazine.