WASHINGTON — The arrival of a new U.S. ambassador to Mexico is usually a routine event. But for the Biden administration, it was a notable victory.

With the Senate’s Aug. 11 confirmation vote, former Senator Ken Salazar of Colorado became the first Biden ambassador to arrive in a foreign capital. And, as of now, the last.

A bitter fight with Senator Ted Cruz, Republican of Texas, over a Russian gas pipeline has created what Biden officials call a personnel crisis, with Mr. Cruz delaying dozens of State Department nominees, including 59 would-be ambassadors, and vowing to block dozens more.

Democrats call Mr. Cruz’s actions an abuse of the nomination process and the latest example of Washington’s eroding political norms. They also say he is endangering national security at a time when only about a quarter of key national security positions have been filled.

While Mr. Cruz cannot entirely block Mr. Biden’s State Department nominees, he has greatly slowed the process by objecting to the Senate’s traditional practice of confirming uncontroversial nominees by “unanimous consent.” His tactic means that each nominee requires hours of Senate floor time while other major priorities, including President Biden’s domestic spending agenda, compete for attention.

“It’s really an undermining of the nation’s national security process,” said Bob Menendez, Democrat of New Jersey and the chairman of the Senate Foreign Relations Committee. “What we have here is an unprecedented, blanketed holding of all nominees — regardless of whether they have anything to do with the policy issues at stake.”

“That is not something I have seen in 30 years of doing foreign policy work” in Congress, he added. “This is unprecedented.”

Only a dozen of Mr. Biden’s State Department nominees have been cleared for a full Senate vote by the committee — in part, Democrats say, because Republicans on the committee are doing their own foot-dragging. Dozens more are expected to be ready for confirmation soon.

Even by the standards of a Senate where political grandstanding is the norm, Democrats say that Mr. Cruz is blatantly exposing his 2024 presidential ambitions by picking a long-running battle with Mr. Biden.

It is one that has attracted an imitator: Senator Josh Hawley, Republican of Missouri, is vowing to block all national security nominees over the Biden administration’s handling of Afghanistan, insisting he will not budge until Secretary of State Antony J. Blinken, Secretary of Defense Lloyd J. Austin III and Jake Sullivan, the national security adviser, resign.

Mr. Cruz and his allies insist he is taking a principled stand on Nord Stream 2, a gas pipeline project from Russia to Germany that has long been an issue of high interest for him.

In mid-May, Mr. Biden waived congressionally imposed sanctions on the project. Critics say the deal will provide President Vladimir V. Putin of Russia leverage over European energy security and deal a blow to Washington’s ally Ukraine, which operates a competing pipeline. But the project is a boon for Germany, and Mr. Biden — pleading that the project was nearly complete and virtually impossible to stop — decided to prioritize relations with Germany, a key European ally, rather than risk a battle with Chancellor Angela Merkel and her soon-to-be-successor.

When that happened, Mr. Cruz accused the Biden administration of showing “weakness” toward Russia. He has since exploited Senate rules to turn confirmation votes — even generally routine ones, for career foreign servants headed for relatively midlevel jobs or low-profile ambassadorships — into hourslong exercises.

“President Biden has insisted on giving a multibillion-dollar gift benefiting Russia, hurting America and hurting our national security interests,” Mr. Cruz said on the Senate floor in August.

“I’ve made clear to every State Department official, to every State Department nominee, that I will place holds on these nominees unless and until the Biden administration follows the law and stops this pipeline and imposes the sanctions,” he added.

Mr. Cruz is not the only reason nearly every foreign ambassadorship and many other State Department positions remain unfilled: The Biden White House was notoriously slow to begin offering foreign policy nominations, exasperating even its Democratic allies.

But unless Mr. Cruz backs down, it could be months before Mr. Biden has his picks in capitals like Beijing, Jerusalem, Cairo and Berlin, and in important policymaking positions at the State Department and the U.S. Agency for International Development.

A minor breakthrough came at the end of September, after Senator Chuck Schumer of New York, the majority leader, saying he would “take the long way,” cleared several hours on the Senate floor to advance some of the State Department nominees. Six were confirmed this past week, most by wide margins, including assistant secretaries of state for European, African, and East Asian and Pacific affairs. But many dozens are still waiting.

State Department officials say the shortage of confirmed senior personnel is straining their ability to conduct diplomacy. They point to the example of Bonnie Jenkins, who was officially nominated in March to be the State Department’s top arms control officer but not confirmed until July 21 — just a few days before she departed for strategic arms talks with the Russians in Geneva.

Mr. Cruz and other Republicans say a 2017 law — the Countering America’s Adversaries Through Sanctions Act, devised to force a reluctant President Donald J. Trump to impose sanctions on Moscow — requires Mr. Biden to penalize Nord Stream 2 AG, the company in charge of the pipeline project, which is a subsidiary of the majority state-owned Russian energy company Gazprom. They say an administration report finding that the company facilitated “deceptive” transactions should trigger the 2017 measure.

Mr. Cruz’s allies say he felt misled by early statements from Mr. Blinken suggesting he would work to stop the Nord Stream project, which helped persuade Mr. Cruz to lift earlier holds on nominees, including Mr. Biden’s pick for director of the C.I.A., William J. Burns.

“This is not something he does lightly or with relish. This is just something he is deeply, deeply concerned about,” said Victoria Coates, a former national security aide to Mr. Cruz who also worked in the Trump White House. “He feels like they lied to him, and they are not understanding how serious this is.”

Democrats say that even if Mr. Cruz is motivated by principle, his reaction is reckless — and wildly out of proportion.

“We can give him the benefit of the doubt that his goal is to micromanage U.S. foreign policy,” said Senator Christopher S. Murphy, Democrat of Connecticut and a member of the Foreign Relations Committee.

“This is not about his objection to Nord Stream 2. This is to get a lot of eyeballs from a fight with President Biden,” Mr. Murphy said. Some Democrats also note that Mr. Cruz has a parochial interest in quashing a foreign energy project that competes with his home state’s oil and gas industry.

Mr. Cruz has not spoken to Mr. Blinken about the matter, though his office has negotiated specific agreements with the State Department. In mid-September, Mr. Cruz allowed unanimous consent votes to confirm three nominees, including assistant secretaries of state overseeing the Western Hemisphere, South and Central Asia, and intelligence.

Mr. Cruz has offered the Biden administration a deal: Impose sanctions on Nord Stream 2 AG, and then waive them if you wish. Those steps would automatically trigger a vote in Congress on whether to block Mr. Biden’s override — one that the president is likely to win, but that would create an unwelcome diversion for the White House, not to mention a platform for Mr. Cruz.

In a statement, Mr. Cruz’s press secretary, Dave Vasquez, said the senator “has worked day in and day out to craft and advance compromises” on the matter, adding that the administration “could get its nominees through tomorrow by simply implementing the law.”

But even under that scenario, Mr. Cruz has promised only to drop his opposition to career foreign service nominees, suggesting that he will continue to deny easy confirmation to political appointees. That category includes former Mayor Rahm Emanuel of Chicago to be ambassador to Japan, former Senator Jeff Flake, a Republican, to be ambassador to Turkey, and the banker and former Obama State Department official Thomas R. Nides to be ambassador to Israel.

Mr. Murphy, who supports changing the Senate’s rules to limit the time that can be devoted to midlevel nominations, called the odds of a Biden reversal on the Nord Stream project, which has been completed but is not yet operational, “negative 75 percent.”

And even if Mr. Cruz were somehow satisfied, there remains Mr. Hawley.

Mr. Menendez suggested that Mr. Schumer could call a weekend Senate session, which would force senators to spend their days off plowing through nominations.

“Ted Cruz and Josh Hawley are not paying any consequences,” Mr. Menendez said. “But when members have to be here on a weekend, voting only on these things that are passing overwhelmingly in a bipartisan vote, I think peer pressure might be brought to bear.”