Kirk was born to a farmer and schoolteacher in Jeffersonville, Ohio. He originally intended to become a foreign correspondent, but fell into educational administration when he served briefly as a high-school principal in New Paris, Ohio during his senior year at college. He graduated from Miami University in 1924, earned a master's degree from Clark University, and studied at the École Libre des Sciences Politiques before completing a Ph.D. at the University of Wisconsin-Madison in 1930. During his studies, he edited his college fraternity's national magazine, The Laurel of Phi Kappa Tau, to earn money for tuition. He married the former Marion Sands, a schoolteacher and daughter of an official of the B&O Railroad, in 1925. They raised one son, John Grayson.
Kirk spent the next decade teaching at the University of Wisconsin-Madison before taking a position as an Associate Professor of Government at Columbia in 1940. During World War II, Kirk began a long association with the U.S. Federal Government when he served in the Security Section of the Department of State's Political Studies Division. Kirk became involved in the formation of the United Nations Security Council, attending the Dumbarton Oaks Conference and the United Nations Conference on International Organization where the United Nations Charter was signed.
Dwight D. Eisenhower, Columbia's then-president, appointed Kirk to the position of University Provost in 1949. In 1951, when Eisenhower took leave to serve as Supreme Commander of NATO, Kirk became acting president of the University. He assumed the presidency in 1953 after Eisenhower was sworn in as President of the United States.
During his tenure at Columbia, he quadrupled the University's endowment, added a dozen new buildings to the Morningside Heights campus, and doubled the University library's holdings.
Kirk's relationship with the student body began to degenerate in the early 1960s as students got caught up in the civil rights and anti-war movements and began to protest openly on campus.
In 1959, Kirk started to pursue the construction of a gymnasium suitable for intercollegiate sports competition. Construction was delayed for several years due to lack of funds, during which time community resentment over the University's crowding out its poorer neighbors festered. When construction began in February, 1968, Harlem community activists and civil rights figures protested vigorously enough for the University to fence off the site and post a police guard.
Also in 1959, Kirk entered Columbia into its relationship with the Institute for Defense Analyses, which would draw much fire from the anti-war movement, particularly the Students for a Democratic Society, nearly a decade later.
In 1967, the University and Kirk came under fire for attempting to patent and promote a "healthier" cigarette filter developed by one of the medical faculty. Questions regarding the filter's effectiveness began to surface just before Kirk was to testify before Congress as to its benefits.
On April 23, 1968, student protesters began what would become an eight-day, five building takeover of campus. Kirk initially agreed to address some of the protesters demands, but ultimately filed trespass charges against them and called in police to clear the occupied buildings. After the incident, Kirk resisted calls for his resignation, but stayed away from graduation and eventually announced his retirement before the start of the next academic year.
After relinquishing the presidency, Kirk served out terms in Council on Foreign Relations, serving as its president until 1971 and the Association of American Universities. Kirk was the recipient of honorary degrees from a number of institutions including: University of Wisconsin-Madison, 1953; Yale University, 1953; Harvard University, 1954; University of North Dakota, 1958; Bates College, LL.D., 1964; Waseda University, 1965.
Kirk died in his sleep at his home in Bronxville, New York in 1997. He is buried next to his wife in Jeffersonville, Ohio.