WASHINGTON — Democrats signaled on Thursday that they would filibuster the nomination of Judge Neil M. Gorsuch, President Trump’s Supreme Court pick, setting up a showdown with Republicans who may be forced to change longstanding rules to seat him on the nation’s highest court.

“He will have to earn 60 votes for confirmation,” Senator Chuck Schumer of New York, the Democratic leader, said on the Senate floor Thursday morning, citing the threshold for breaking a filibuster on the selection. “My vote will be no.”

The announcement came one day after Judge Gorsuch completed his second day of testimony before the Senate Judiciary Committee, emerging largely unscathed amid a series of bland deflections and folksy digressions.

Many Democrats are facing dual pressures as they make their decisions: The party’s progressive base has pressed them to oppose Mr. Trump at every turn, and many are still seething over the treatment last year of Judge Merrick B. Garland, President Obama’s nominee, whom Republicans refused to consider in an election year.

But several lawmakers face re-election races next year in states that Mr. Trump won, compelling some to weigh supporting Judge Gorsuch. Still, based on interviews and internal discussions, Judge Gorsuch appears to be short — at least for now — of the eight Senate votes he must earn from the Democratic caucus to reach 60 votes. (Republicans hold a majority with 52 seats.)

Also on Thursday, Senator Bob Casey, Democrat of Pennsylvania, who is up for re-election next year, said he would vote against Judge Gorsuch.

“I have serious concerns about Judge Gorsuch’s rigid and restrictive judicial philosophy,” he said, suggesting that the nominee “employs the narrowest possible reading of federal law and exercises extreme skepticism, even hostility, toward executive branch agencies.”

Both Mr. Schumer and Mr. Casey echoed longstanding Democratic attacks on Judge Gorsuch: that his decisions tend to favor the powerful.

“His career and judicial record suggest not a neutral legal mind but someone with a deep-seated conservative ideology,” Mr. Schumer said.

Republican leaders have signaled an openness to changing longstanding rules regarding the filibuster and confirming Judge Gorsuch on a simple majority vote. And Mr. Trump has urged them to pursue this so-called nuclear option if necessary.

On Thursday, Mr. Schumer asked his colleagues to avoid the temptation.

“If this nominee cannot earn 60 votes — a bar met by each of President Obama’s nominees and George Bush’s last two nominees — the answer isn’t to change the rules,” he said. “It’s to change the nominee.”