Born in New York City, Friedman graduated from Brooklyn College in 1936 and obtained his PhD in 1940 from Johns Hopkins University, Baltimore. Shortly afterwards he joined the US Naval Research Laboratory in Washington where he spent his whole career, being appointed in 1958 as superintendent of the atmosphere and astrophysics division and in 1963 superintendent of the space science division. Also in 1963 he became chief scientist at the E. O. Hulbert Center for Space Research. In addition he served as adjunct professor at the universities of Maryland and Pennsylvania.

Friedman was a pioneer in both rocket astronomy and in the study of the x-ray sky. The two went hand in hand in the early days of x-ray astronomy for without rockets it would have been impossible to detect any significant x-ray activity in space since x-rays are absorbed by the Earth's atmosphere. Solar x-rays were detected as early as 1948 by T. R. Burnright and were systematically investigated from 1949 by Friedman and his colleagues, who observed x-ray activity throughout a full solar cycle of 11 years. Friedman also studied ultraviolet radiation from the Sun and in 1960 produced the first x-ray and ultraviolet photographs of the Sun.

X-ray astronomy really came of age in 1962 when nonsolar x-ray activity was first discovered by Bruno Rossi. The x-rays came from a source in Scorpio, since named Sco X–1. A second source was discovered in 1963 in Taurus and named Tau X–1. Friedman made the first attempt to locate accurately an x-ray source two years later: when the Moon passed in front of the Crab nebula, a luminous supernova remnant in Taurus, the x-ray activity of Tau X–1 was found to fade out gradually. Tau X–1 was therefore identified with the Crab nebula and seemed to be a source about a light-year across lying in the center of the nebula.

Since then satellites carrying x-ray equipment have been launched, including Uhuru in 1970 and the Einstein Observatory in 1978. These have enormously extended the scope of x-ray astronomy and shown its value in the search for neutron stars and black holes.