He was born in Trumbull County, Ohio on March 5, 1830. He passed the bar and set up practice in both Clearfield, Pennsylvania and Falls City, Nebraska from 1853 to 1858, and from 1862 to 1863. He was a member of the Nebraska Territorial Council from 1858 to 1862.
From 1863 to 1867 Dundy served on the Territorial Supreme Court. When Nebraska became a state on April 4, 1868, he was nominated by President Andrew Johnson to a new seat as a judge on the United States District Court for the District of Nebraska. He was confirmed by the United States Senate on April 9, 1868. In April, 1879 Judge Dundy ruled that Standing Bear and other Ponca were "people" who able to bring petitions for habeas corpus, and that Indians who had severed their relationships with their tribes could not be ordered to a reservation against their will. The next year, Dundy was part of the lower court panel that heard Elk v. Wilkins, which asserted that Indians who had left their tribes and submitted to the jurisdiction of the United States were American citizens. In 1884, the U.S. Supreme Court rejected Elk's petition, holding that Indians born in tribal relations in the United States could only become citizens under specific federal laws. It was not until 1924 that all Indians born in the United States were declared citizens with the passing of the Indian Citizenship Act of 1924.
He served until October 28, 1896 when he died in Omaha, Nebraska.
Judge Dundy's son Elmer Scipio Dundy Jr., better known as “Skip” Dundy, was born in Omaha in 1862. Skip Dundy grew up to become a promoter on Coney Island, due to in part the stories told by Buffalo Bill Cody who was a familiar visitor in the Dundy home.