Wildt was born in Munich, Germany, and studied at the University of Berlin, obtaining his PhD there in 1927. After a period of teaching at Göttingen he emigrated to America in 1934 and worked at various institutions including the Mount Wilson Observatory (1935–36), Princeton University (1937–42), and the University of Virginia (1942–46). In 1948 he took up an appointment at Yale University where he served as professor of astrophysics from 1957 until his retirement in 1973.

Wildt worked in stellar spectroscopy, theoretical astrophysics, and geochemistry, but his main interest lay in planetary studies. Since much of his work was done before the start of the space program it is not surprising that many of his speculations have since been rejected. Thus his claim that the Venusian clouds contained formaldehyde (CH₂O) formed under the influence of ultraviolet rays has not been confirmed by space probes.

He was, however, successful in his identification in 1932 of certain absorption bands observed by Vesto Slipher in the spectrum of Jupiter as being due to ammonia and methane. He also, in 1943, proposed a model of the constitution of Jupiter based on its density of 1.3 arguing that it consisted of a large metallic rocky core surrounded by ice and compressed hydrogen above which lies a thick layer of atmosphere. Saturn, Uranus, and Neptune were thought to have a similar constitution, but more recent models, together with the Pioneer and Voyager space flights, have suggested that the giant planets are made mainly of hydrogen in various forms with much smaller cores than Wildt proposed.