He was educated there and at the Friedrich-Wilhelms-Universität in Berlin, graduating in 1906.
He wrote a doctoral dissertation there on partial differential equations and the Plateau problem, in 1910, supervised by H. A. Schwarz. In 1911 he moved to Munich. In the following years he published on projective geometry, iterative methods, and approximation theory.
In 1914 he took a teaching position in a school near Heppenheim, and a year later another in Hochwaldhausen. He became a German citizen in 1919. He shortly began publishing mathematics again.
The couple moved to Göttingen in 1921, and Müntz became involved in editorial, reviewing and translation work, as well as research. At this period, and from 1924 in Berlin, he was unsuccessfully trying to get an academic position, hampered because he had not habilitated. In 1927 he worked closely with Albert Einstein.
In 1929 he took a professorial position at the Leningrad State University, where he was active in teaching, research, administration and as an editor of Lyapunov. In 1932 he was an official Soviet delegate to the International Congress of Mathematicians, with N. G. Chebotarev, P. S. Alexandrov, and E. Y. Kolman, an ideological Marxist. While in Russia, Müntz was influential in helping several other mathematicians escape from the Nazis.
In 1937 Müntz, who remained a German citizen, was expelled from the USSR. He went to Sweden, where he supported himself by teaching. He obtained Swedish citizenship in 1953. In 1907 a book Wir Juden was published in Berlin.
He was a correspondent of Martin Buber, and wrote much for Buber's journal Der Jude.