I am surprised nobody brought this up:
Supreme Court Justice Lewis Powell was considered a moderate, often referred to as a "swing vote" in close decisions. Even before his expected retirement on June 27, 1987, Senate Democrats had asked liberal leaders to form a "solid phalanx" to oppose whomever President Ronald Reagan nominated to replace Powell, assuming that it would tilt the court rightward. Democrats warned Reagan there would be a fight over the nomination if Bork were to be the nominee.
Reagan nominated Bork for the seat on July 1, 1987. Bork had long been interested in the position; in 1973, President Nixon promised him the next seat on the Supreme Court following Bork's compliance in the controversial Saturday Night Massacre. Nixon was unable to carry out the promise before his resignation. 
Within 45 minutes of Bork's nomination to the Court, Senator Ted Kennedy (D-MA) took to the Senate floor with a strong condemnation of Bork in a nationally televised speech, declaring,
Robert Bork's America is a land in which women would be forced into back-alley abortions, blacks would sit at segregated lunch counters, rogue police could break down citizens' doors in midnight raids, schoolchildren could not be taught about evolution, writers and artists could be censored at the whim of the Government, and the doors of the Federal courts would be shut on the fingers of millions of citizens.
On July 5th, NAACP executive director Benjamin Hooks described their position on the Bork nomination: "We will fight it all the way - until hell freezes over, and then we'll skate across on the ice." A brief was prepared for Joe Biden, head of the Senate Judiciary Committee, called the Biden Report. Bork later said in his book The Tempting of America that the report "so thoroughly misrepresented a plain record that it easily qualifies as world class in the category of scurrility". TV ads produced by People For the American Way and narrated by Gregory Peck attacked Bork as an extremist, and Kennedy's speech successfully fueled widespread public skepticism of Bork's nomination. The rapid response of Kennedy's "Robert Bork's America" speech stunned the Reagan White House; though conservatives considered Kennedy's accusations slanderous, the attacks went unanswered for two and a half months.
A hotly contested United States Senate debate over Bork's nomination ensued, partly fueled by strong opposition by civil and women's rights groups concerned with Bork's stated desire to roll back civil rights decisions of the Warren and Burger courts, and his opposition to the right of the Federal government to impose standards of voting fairness upon the states. Bork is one of only three Supreme Court nominees to ever be opposed by the ACLU. Bork was also criticized for being an "advocate of disproportionate powers for the executive branch of Government, almost executive supremacy", as demonstrated by his role in the Saturday Night Massacre.
During debate over his nomination, Bork's video rental history was leaked to the press, which led to the enactment of the 1988 Video Privacy Protection Act. His video rental history was unremarkable, and included such harmless titles as A Day at the Races, Ruthless People, and The Man Who Knew Too Much. The list of rentals was originally printed by Washington D.C.'s City Paper.
To pro-choice legal groups, Bork's originalist views and his belief that the Constitution does not contain a general "right to privacy" were viewed as a clear signal that, should he be named to the Supreme Court, he would vote to reverse the Court's 1973 decision in Roe v. Wade. These groups also claimed that Bork's second marriage to a former Roman Catholic nun would allow her to influence his decisions on the abortion issue. Accordingly, a large number of left-wing groups mobilized to press for Bork's rejection, and his confirmation hearings became an intensely partisan battle. Bork himself became a Catholic in 2003, and his replacement (Justice Kennedy) was a Catholic, as well as 4 of the next 6 Justices to join the court. Bork was faulted for his bluntness before the committee, including his criticism of the reasoning underlying Roe v. Wade. Simultaneously, however, his supporters expressed frustration that some of Bork's most controversial and conservative views, including those on the scope of the First Amendment and the Equal Protection clause of the Fourteenth Amendment, as expressed in his writings and past opinions, had been suddenly moderated for his testimony before the Committee.
As Chairman of the Judiciary Committee, Senator Joe Biden presided over Bork's hearing. Biden stated his opposition to Bork soon after the nomination, reversing an approval in an interview of a hypothetical Bork nomination he had made the previous year and angering conservatives who thought he could not conduct the hearings dispassionately. At the close of the hearings, Biden won praise for conducting the proceedings fairly and with good humor and courage, as his 1988 presidential campaign collapsed in the middle of the hearings. Rejecting some of the arguments that other Bork opponents were making, Biden framed his discussion around the belief that the Constitution provides rights to liberty and privacy that extend beyond those explicitly enumerated in the text, and that Bork's strong originalism was ideologically incompatible with that view.
On October 6, Bork's nomination was rejected in the committee by a 9–5 vote. Since a committee rejection made a rejection by the full (Democratic) Senate extremely likely, Bork was widely expected to concede defeat and withdraw his name from a floor consideration. However, three days later, Bork announced his belief that
There should be a full debate and a final Senate decision. In deciding on this course, I harbor no illusions. But a crucial principle is at stake. That principle is the way we select the men and women who guard the liberties of all the American people. That should not be done through public campaigns of distortion. If I withdraw now, that campaign would be seen as a success, and it would be mounted against future nominees. For the sake of the Federal judiciary and the American people, that must not happen. The deliberative process must be restored.
Faced with certain defeat, Bork's political support fell silent; Bork would even express disappointment with Reagan's tepid continued endorsement. On October 23, 1987, the Senate rejected Bork's confirmation, with 42 Senators voting in favor and 58 voting against. Senators David Boren (D-OK) and Ernest Hollings (D-SC) voted in favor, with Senators John Chafee (R-RI), Bob Packwood (R-OR), Arlen Specter (R-PA), Robert Stafford (R-VT), John Warner (R-VA) and Lowell P. Weicker, Jr. (R-CT) all voting nay. The vacant seat on the court to which Bork was nominated eventually went to Judge Anthony Kennedy.