He was born in Munich, and was a pupil of Carl Friedrich Philipp von Martius. Having quarrelled with his family, he sailed to Java and entered the Dutch service, in which he stayed for several years. In 1864 he was induced by Thomas Anderson, who was visiting the Dutch colonies, to return with him to Calcutta as curator of the herbarium, a post he held till his death, to the great advantage of Indian botany.
Kurz was often sent on botanical missions. He explored Burma and Pegu, and spent three months in the Andaman Islands, of which he gave an exhaustive report in 1870. He died in Penang on 15 January 1878.