Director of the Royal Gallery in Cassel, 1910-1924. Gronau was born to a Jewish family. He studied in Bonn and Berlin, writing his thesis at the latter institution on Urspergensis Burchardus (d. 1230) and the Chronicon in 1890. From the first, his personal research interest was in Venetian art. Gronau purchased an Italian villa in the 1890s, San Domenico, in Fiesole where he lived until 1910. There he published articles beginning in 1894 and issued his book on Titian in 1900. In 1902 he published a book on Raphael, followed by an English book on Leonardo, 1903, a translation of the Titian work in 1904, and a book on Correggio in 1907. His book, Die künstlerfamilie Bellini (1909) became one of the early required texts to be listed in the course catalog for the art history classes of Princeton University in the United States. The following year, 1910, he left Italy to become Director of the Gemäldegalerie in Cassel, Germany. During World War I, one of his sons was killed in combat. He retired from the Gemäldegalerie in 1924 continuing a long association with the Kunsthistorisches Institut in Florence, returning to live at San Domenico in 1929. In 1930, Gronau published his Bellini chronology, a work which met with a great deal of dissent. A gifted archivist-researcher in the German tradition, he spent his final years compiling Documenti artistici Urbinati, published in 1936, an annotated index of the archives of Urbino relating to artists. A book on the documents of Titian was never realized. He died at San Domenico in 1938. His son was the art historian Hans-D. Gronau.
Robert Witt estimated that Gronau was one of the first to understand the importance of Giovanni Bellini and to establish a chronology of that artist's work, which subsequent scholars used to modify or build upon. Roger Fry reviewing Gronau's book on Raphael, called Gronau "an unusually gifted and scientific observer." He collaborated with the Dresden museum director Woldemar von Seidlitz on Leonardo scholarship. He worked with Herbert Cook on establishing the date of Titian's birth.