Cox was the son of Archibald and Frances Perkins Cox. A native of Plainfield, New Jersey, he attended the Wardlaw-Hartridge School, then called Wardlaw Country Day, and St. Paul's School. Cox graduated from Harvard College during 1934 and from Harvard Law School during 1937 where he was a member of Phi delta phi legal fraternity. He was a clerk for U. S. Judge Learned Hand of the U.S. Court of Appeals for the Second Circuit. After his clerkship, he joined the Boston law firm of Ropes, Gray, Best, Coolidge and Rugg, now known as Ropes & Gray. During World War II, he was appointed to the National Defense Board, and then to the Office of the Solicitor General.

After the end of World War II, Cox joined the faculty of Harvard University, where he taught courses in torts and in administrative, constitutional, and labor law. During the 1950s, he became an informal adviser and speech-writer for John F. Kennedy, who was then a U.S. senator from Massachusetts. Cox assisted Kennedy's campaign for President during 1960. During 1961, Cox joined the Kennedy administration as solicitor general. At a time when civil rights protesters were routinely chased with dogs and clubbed, he often appeared before the Supreme Court in support of their cause. Among the cases he was involved with were Baker v. Carr, which set the constitutional standards for reapportionment; Heart of Atlanta Motel v. United States, which set a precedent by recognizing the Constitution's authorization for federal laws requiring desegregation of public accommodations for African-Americans; and South Carolina v. Katzenbach, which upheld the Voting Rights Act. During 1965, he returned to Harvard's law school.

On May 19, 1973, Cox took a leave of absence from Harvard Law School to accept appointment as the first Watergate special prosecutor. Cox's appointment was a major condition set by the management of the U. S. Senate for the confirmation of Elliot Richardson as the new attorney general of the United States, succeeding Richard G. Kleindienst, who had resigned during the spring of 1973, as a result of the Watergate scandal. That summer, Cox learned with the rest of America about the secret taping system installed in the White House on orders from President Richard M. Nixon. During the next few months, Cox, the Senate Watergate committee, and U.S. District Judge John J. Sirica struggled with the Nixon Administration over whether Nixon could be compelled to yield those tapes in response to a grand jury subpoena.When Sirica ordered Nixon to comply with the committee's and Cox's demands, the President offered Cox a compromise: instead of producing the tapes, he would allow the Senator John Stennis (Democrat -- Mississippi) to listen to the tapes, with the help of a transcript prepared for him by the White House, and Stennis would then prepare summaries of the tapes' contents. Cox rejected this compromise on Friday, October 19, 1973. On Saturday, October 20, 1973, Cox had a press conference to explain his decision. That evening, in an event dubbed the Saturday Night Massacre by journalists, President Nixon ordered Attorney General Elliot Richardson to dismiss Cox. Rather than comply with this order, Attorney General Richardson resigned, leaving his second-in-command, Deputy Attorney General William Ruckelshaus in charge of the Justice Department. Ruckelshaus likewise refused to dismiss Cox, and he, too, resigned. These resignations left Solicitor General Robert Bork as the highest-ranking member of the Justice Department; insisting that he believed the decision unwise but also that somebody had to obey the president's orders, Bork dismissed Cox. Upon being dismissed, Cox stated, "whether ours shall be a government of laws and not of men is now for Congress and ultimately the American people to decide."

The dismissal of Cox suggested the use of independent counsels — prosecutors specifically appointed to investigate official misconduct; ultimately, Congress enacted a law to provide for a procedure appointing independent counsels, a statute that the U.S. Supreme Court upheld during 1986. This statute, which had an expiration date inserted on its original enactment, expired without renewal.

Ultimately, on 8 August 1974, after the U.S. Supreme Court voted by 8 to 0 to reject Nixon's claims of executive privilege and release the tapes (with then Associate Justice William H. Rehnquist recusing himself because, as an assistant attorney general during Nixon's first term, he had taken part in internal executive-branch discussions of the scope of executive privilege), Nixon announced his decision to resign as president.

After Nixon's resignation, Cox became chairman of Common Cause, a major public-interest advocacy organization. He also became the founding chairman of the Health Effects Institute.

Cox also returned to Harvard Law School, where he taught constitutional law and a seminar on the First Amendment for many years. Before he had gone to Washington during 1973, he had a reputation as a tough and sometimes harsh teacher, but after his return, he had a reputation as a humorous, considerate, and gentle teacher who won the admiration and affectionate regard of his students. After he retired from Harvard, he received a special appointment to the faculty of Boston University School of Law. In 1974 he spent a year at the University of Cambridge as the Pitt Professor of American History and Institutions.

Cox also continued his career as an expert appellate advocate. During 1976, Cox argued Buckley v. Valeo before the Supreme Court; at issue in this case was the constitutionality of post-Watergate legislation establishing public financing for presidential election campaigns. The Court upheld most of the provisions of the campaign finance law, giving Cox a significant victory. During 1977 and 1978, Cox also argued the case of Regents of the University of California v. Bakke before the Court, defending the University of California at Davis medical school's affirmative action system of admissions against constitutional challenge. The Justices divided, with four voting to end the system as invalid under the Civil Rights Act of 1964 without any need to reference the constitutional issue, and four voting to uphold affirmative action as constitutional; Justice Lewis H. Powell made the deciding vote, referencing the constitutional issue and holding that in some cases race could be deemed a valid factor in admissions to institutions of higher education.

During 1979, when a vacancy opened on the U.S. Court of Appeals for the First Circuit (which includes Cox's home state of Massachusetts), Senator Edward M. Kennedy proposed Cox for the vacancy. This proposal from the senior senator of the state most affected by the choice of judge ordinarily would have won Cox the appointment, but the administration of President Jimmy Carter resisted the choice since Cox had not supported Carter for President, and ultimately Cox was not appointed to the vacancy.

Among Professor Cox's many honors was his being made, during 1991, an honorary member of the Order of the Coif. Professor Cox also received the Paul Douglas Ethics in Government Award and the Thomas "Tip" O'Neill Citizenship Award.

On January 8, 2001, Cox was presented with the Presidential Citizens Medal by President William Clinton.

Cox's many books include The Warren Court: Constitutional Decisionmaking as an Instrument of Reform (Harvard University Press, 1969); The Role of the Supreme Court in American Government (Oxford University Press, 1976), Freedom of Expression (Harvard University Press, 1982), and The Court and the Constitution (Houghton Mifflin, 1987). He also wrote many leading law review articles and was co-editor of the leading casebook on labor law.

Cox died at his home in Brooksville, Maine, of natural causes, on the same day as Sam Dash, chief counsel to the Senate Select Committee to Investigate Campaign Practices during the Watergate scandal.

The New York Times wrote, "a gaunt 6-footer who wore three-piece suits, Mr. Cox was often described as 'ramrod straight,' not only because of his bearing but also because of his personality."

Cox was the great-grandson of William M. Evarts, who defended President Andrew Johnson during his impeachment hearing and became Secretary of State in the Hayes administration. He was also a direct descendant of Roger Sherman, a Connecticut signer of the Declaration of Independence.

A partial list of Cox's books:

• The Warren Court: Constitutional Decision as an Instrument of Reform (1968) ISBN:
• Freedom of Expression (1981) ISBN 0735102368
• Law and the National Labor Policy (1983) ISBN 0313237948
• The Role of the Supreme Court in American Government (1976) ISBN 0198274114
• The Court and the Constitution (1987) ISBN 039548071X

Taught current Marquette University Law Professor Ed Fallone and New York Law School distinguished adjunct professor Richard B. Bernstein.