Arizona Republican Sen. Jeff Flake said Wednesday that he would not vote for any more of President Trump’s judicial nominees until the Senate votes on a bill to prevent special counsel Robert S. Mueller III from being fired — a pledge that could complicate Republicans’ hope to confirm dozens of conservative judges before the end of the year.

Flake’s warning will likely force Republicans — who hold 51 seats in the Senate — to rely on Vice President Pence to confirm any of the 32 judicial nominees pending before the full Senate, as Democrats have little incentive to support those who Flake has committed to oppose. It also means that Republicans will likely have to go around the Senate Judiciary Committee, where Flake is a member and the GOP has only a one-seat majority, to advance any of the 21 nominees waiting for that panel’s endorsement. That also will require Pence’s tie-breaking vote.

Flake issued his threat after Senate Majority Leader Mitch McConnell (R-Ky.) blocked Flake and Sen. Christopher A. Coons (D-Del.) from holding a vote on the bill, which would give any fired special counsel the ability to swiftly challenge their termination before a panel of three federal judges. Most Republicans — including co-authors Sens. Thom Tillis (R-N.C.) and Lindsey O. Graham (R-S.C.) — have argued that the bill is unnecessary because Trump would never dare fire Mueller, whose ongoing probe of Russian interference in the 2016 election has sought to learn whether anyone in Trump’s campaign conspired with those efforts.

Flake challenged that rationale, given Trump’s recent decision to appoint Matthew G. Whitaker as acting attorney general and give him oversight of the Mueller probe.

Whitaker has made past statements that are critical of the investigation. Flake said he believes Whitaker should recuse himself from the Russia probe, letting deputy attorney general Rod J. Rosenstein reassume authority over it.

“The president now has this investigation in his sights and we all know it,” Flake said on the Senate floor Wednesday, emphatically asking his colleagues why they were content to allow a potential constitutional crisis to unfold.

“Why? Why do we do this? To protect a man seemingly who is so incurious about what Russia did during the 2016 elections?” Flake said, referring to Trump. “Why do we do that? Do we have no more institutional pride here?”

Flake, who is retiring at the end of the year, has become one of Trump’s most vocal Republican critics in Congress. His pledge to vote against Trump’s judicial nominees unless the special counsel bill is voted on is the most concrete step he has yet taken to try to deny Trump something tangible, instead of just decry his actions on the Senate floor.

“You use what leverage you have,” Flake told reporters, explaining why he made his threat. “This is a priority now.”

By himself, Flake cannot completely foul up GOP leaders’ plans to confirm more conservative judges this year. But he and Coons said they hope to convince other Republicans to join their effort. If one more Republican does, they and the Democrats would be able to prevent Trump from getting any additional judicial nominees confirmed in 2018 — a move that Flake guessed would at least send a strong message about the importance of the special counsel bill.

“We are confident [the bill] would get 60 votes if given a vote,” Coons told reporters. “It is time for us to move from speech to action.”

One senator that Coons and Flake may target in the days ahead is Sen. Susan Collins (R-Maine). Last week, she said that she thought the Senate should vote on the bill.

“Senate debate and passage of this bill would send a powerful message that Mr. Mueller must be able to complete his work unimpeded,” Collins said in a statement, in which she also said she was “concerned” about Whitaker’s views on Mueller’s probe.

Even if senators succeed in passing the bill, its fate in the House is uncertain. Though House Speaker Paul D. Ryan (R-Wis.) has defended Mueller’s right to conduct his probe unimpeded, he has also determined that a bill to protect him is not necessary. Still more Republicans have charged that the legislation is unconstitutional, as it allows the courts to review the president’s hiring and firing decisions, which are considered his purview.

Given those circumstances, the best chance of passing the bill this year is if senators manage to convince leaders to include it in must-pass legislation, such as the spending bill lawmakers need to approve early next month.

In the meantime, it is unclear whether Whitaker will indulge his previously stated desire to constrain Mueller’s probe, or permit it to continue unrestricted.

This week, the state of Maryland filed a lawsuit challenging the constitutionality of Whitaker’s appointment. Some legal experts have suggested that Whitaker, who was former attorney general Jeff Sessions’s chief of staff, is ineligible for the top Justice Department job, even on an acting basis, because he was not confirmed by the Senate.

Earlier Wednesday, the Justice Department issued a memo defending the legality of Whitaker’s appointment.

This is not the first time Flake has broken from his party to team up with Coons. Last month, the duo joined forces to push the Senate to demand an FBI investigation into sexual assault allegations made against Brett M. Kavanaugh before voting on his nomination to the Supreme Court. In that instance, they succeeded.

Devlin Barrett contributed to this report.