The Supreme Court nomination of Judge Robert H. Bork appeared headed for defeat yesterday after Senate Majority Leader Robert C. Byrd (D-W.Va.) and three other senators, including the third-ranking member of the Senate Republican leadership, announced that they will vote against Bork's confirmation.

Byrd called on President Reagan to withdraw the nomination "and save Judge Bork, the court and the country the pain" of prolonging the confirmation battle.

"What the White House should do is to recognize that this nomination is going down," Byrd said. "It can't go up, it's going down . . . . The nomination is doomed."

But Reagan refused to yield. When a television reporter shouted at the president after a White House Rose Garden ceremony that it "looks like the Bork nomination will fail in committee," Reagan fired back, "Over my dead body."

Administration officials acknowledged, however, that the tide seemed to be running against Bork, and a White House count showed steady erosion among potential Senate supporters. Last week, White House officials said they needed "eight or nine" votes from a list of 18 undecided senators to win. Two of these senators then came out against Bork.

Yesterday, two more undecided senators -- Dennis DeConcini (D-Ariz.) and John H. Chafee (R-R.I.) -- came out against the nomination. "We still need eight or nine votes and the list from which to get them is shrinking," a White House official said.

Most of the remaining undecided senators are southern Democrats, and the White House considers the decision of Sen. Howell Heflin (D-Ala.), the only uncommitted member of the Judiciary Committee, as the key to winning them. How Heflin votes today when the panel considers the nomination "will send a big signal to other southern Democrats," the official said.

But the announcements yesterday by panel members Byrd and DeConcini meant that a majority of the committee will vote against confirmation. Senate sources said the committee is expected to report the nomination to the Senate floor with an unfavorable recommendation, probably by a 9-to-5 vote if Heflin joins the opposition to Bork.

The announcement by Chafee, chairman of the Senate Republican Conference, and Lowell P. Weicker Jr. (R-Conn.), who was not expected to support Bork, brought the number of GOP senators who oppose Bork to four.

In a statement in Providence, Chafee said Bork's pledge that he would not seek to overturn Supreme Court decisions that he has harshly criticized in the past "has left me extremely uncomfortable."

Byrd said that if Reagan refuses to withdraw the nomination, he will bring Bork's name up for debate on the Senate floor "rather quickly," probably next week. He said there is still time for the Senate to confirm a new nominee to succeed retired justice Lewis F. Powell Jr. this year, but only if the Reagan administration acts quickly to withdraw Bork's name and make a new nomination.

Last week, Byrd said he wanted the Judiciary Committee to send the nomination to the Senate floor with no recommendation and would oppose attempts to attach either a favorable or unfavorable report. He said he changed his mind because "I'm convinced it's doomed and further delay simply contributes to further divisiveness around the country, further unease."

Byrd complimented Bork for his performance during five days of testimony before the Judiciary Committee and said the White House had been warned that his nomination would set off a bitter political struggle.

"I really feel sorry for the man because I don't think he should have been put through that misery," he said.

White House spokesman Marlin Fitzwater insisted that Reagan was not backing away from Bork in any way and was not considering alternative nominations. But White House activity for Bork did not appear to match the claims of Reagan's intense interest in the nomination.

On Friday, Fitzwater had said that Reagan would engage in a public event for Bork each day this week. Yesterday, however, Reagan brushed aside all questions about Bork in one photo-taking session, gave only his brief answer after the Rose Garden ceremony and limited his pro-Bork activity the rest of the day to three telephone calls to senators, officials said.

These officials said it also was unlikely that the president would make a nationally televised speech on Bork's behalf this week, one of the proposals that had been considered by his strategists.

"The president does not want to withdraw the nomination and is convinced that Bob Bork has been the victim of a campaign of distortion," said one official. "He will keep fighting. He knows it is an uphill fight."

Some officials believe that Bork might decide on his own to withdraw if the vote in the Judiciary Committee goes against him. But White House officials insisted yesterday that neither Reagan nor his aides will ask Bork to step aside.

Leonard Garment, who has been assisting Bork in the confirmation battle, said yesterday, "He will never, never, never withdraw . . . . He will fight this to the end."

White House chief of staff Howard H. Baker Jr., a favorite target of conservatives, has been accused by some of them of not doing enough to advance the Bork nomination. An official said that Baker did not consider this criticism valid but was "highly conscious" of it and would make no move "that could in any way be interpreted as exerting pressure on Bork to withdraw."

Attorney General Edwin Meese III has strongly urged that the administration "stand with Bork until there is an up-or-down vote in the Senate," this official said.

The 14 remaining undecided senators on the White House list include 12 Democrats, all but three from the South, and two Republicans. The White House needs votes from at least eight of them to reach 50 votes, in which case Vice President Bush could break a tie in Bork's favor.

The Democrats are Sen. William Proxmire (Wis.), Lawton Chiles and Bob Graham of Florida, Sam Nunn and Wyche Fowler Jr. of Georgia, Heflin and Richard C. Shelby of Alabama, John B. Breaux (La.), Wendell H. Ford (Ky.) and J. James Exon (Neb.), who suggested last week that the White House ought to consider withdrawing Bork. The Republicans are William S. Cohen (Maine) and Robert T. Stafford (Vt.).

Nunn, a key southern Democrat, met in Atlanta yesterday with black civil rights leaders who vehemently oppose Bork. Nunn said after the meeting that "we cannot go backwards in civil rights and other areas where we've made huge gains in the country" and suggested that he is troubled by Bork's views on privacy.

Senate Majority Whip Alan Cranston (D-Calif.) said his latest count showed 52 votes against confirmation, 42 for confirmation and six undecided. Byrd, who had been considered undecided in Cranston's count, raised the estimated opposition to 53 votes. Cranston said he expected 55 or 56 senators to vote against confirmation if the issue comes to a vote on the Senate floor.

DeConcini, who had been considered a key swing vote on the Judiciary Committee, said he urged the White House to withdraw the nomination yesterday. "Someone has to tell this president that this nomination isn't going to go," he said.

DeConcini said Bork had been unfairly depicted by his most extreme critics. But noting that he has two daughters, DeConcini said he remained concerned about Bork's views on the application of equal protection guarantees to women and his rejection of a constitutional right to privacy.

"He is not a racist or a sexist, but he is not a mainstream conservative," DeConcini said.

Bork's only endorsement yesterday came from Sen. Nancy Landon Kassebaum (R-Kan.), who has supported the nomination from the outset. "It's a judgment call, but the vitality he would bring, the dedication and integrity he would bring is important to the Supreme Court," she said.

As the Bork nomination was crumbling, "We the People," an organization created by some of Reagan's friends to campaign for confirmation, announced that it had placed a full-page advertisement in today's editions of USA Today. The ad charged that Judiciary Committee Chairman Joseph R. Biden Jr. (D-Del.) had "manipulated" the Bork hearings "to save his political life" and made personal attacks on committee Democrats Edward M. Kennedy (Mass.), Howard M. Metzenbaum (Ohio) and Patrick J. Leahy (Vt.).

Staff writer Al Kamen contributed to this report.