Born in 1909 in Brooklyn, Greenspun was raised to be proud of his Jewish heritage and to stand up in the face of oppression. He spent most of World War II as an ordinance officer, but a brief stint for a military newspaper led to his opening of the entertainment rag, "Las Vegas Life," with a college buddy. After "Life" went kaput, Greenspun found a job at the Flamingo, working as a publicist for "Bugsy" Siegel.

In 1950, he purchased a small, tri-weekly paper, refashioned it as a daily called the Las Vegas Sun, and set about changing the way business was conducted in Sin City. The Sun quickly became a bastion of contrary views and a vital vehicle for reform in a highly corrupt environment. Greenspun took on U.S. Senator Patrick McCarran, ruthlessly attacking him and his associates, including the editor of the rival Las Vegas Review-Journal and Senator Eugene McCarthy. One of the first journalists to disapprove of McCarthy's red-baiting tactics, Greenspun piled insult after insult on the senator, eventually earning himself an indictment for his rowdy opinions.

Though he often played a significant role in political campaigns, when Greenspun entered the ring himself, for the 1962 gubernatorial candidacy, he was defeated in the primaries. But his paper continued to grow in influence, eventually challenging the Review-Journal for top position. A suspicious fire at the Sun offices in 1963 put a halt to the paper's march, eventually forcing Greenspun to operate under a joint operating agreement with the Las Vegas Review-Journal.

Before his death in 1989, Greenspun operated radio station KRAM and founded KLAS-TV (which he eventually sold to Howard Hughes), owned cable television franchise rights for most of Las Vegas, and was behind the development of Green Valley. The Sun, under the operating agreement developed by Greenspun, remains editorially independent to this day.