As President Trump wrapped up a prime-time address in which he rattled off ominous and at times outlandish assertions about U.S. election vulnerabilities, he repeated his demand that Congress pass strict voting restrictions that have been stalled in the Senate.

That plea, which Mr. Trump implied was the reason behind the Thursday night speech, appeared to have landed with a whimper among Republicans on Capitol Hill and in campaigns around the country. Most had little to say in its aftermath about an issue the president called among the most urgent facing the nation.

Democrats immediately seized on Mr. Trump’s address with a frenzy of talking points to accuse him of trying to undermine the upcoming midterm elections. But the speech drew a muted response from members of the president’s party. He has spent months trying to pressure them into action on the election overhaul measure, which would require proof of citizenship to register and photo ID to vote and would severely limit voting by mail.

Though Mr. Trump’s most devoted loyalists in Congress quickly repeated his call to pass the bill known as the SAVE America Act — demands they have been making for months — most other Republicans stayed quiet, and at least one skeptic indicated that the speech had not moved the needle.

“If it was meant to influence people like me, it didn’t get me there,” Senator Lisa Murkowski, an Alaska Republican who has publicly opposed the measure, said in an interview on Friday. “What we heard last night was there are foreign interests trying to influence us, but they weren’t effective at basically hacking into our system. I don’t think there was much last night that would change views on whether we should pass the SAVE Act.”

Broadly, it was not clear that Mr. Trump’s message was resonating beyond the far right of the Republican Party that has long unquestioningly echoed his baseless and unfounded claims of widespread voter fraud.

Republicans on the campaign trail did not seize on Mr. Trump’s remarks, instead focusing on issues they perceived to be of greater import. And conservative media outlets found other stories as compelling, if not more so, as Mr. Trump’s remarks about election security.

On Friday morning, “Fox & Friends,” the program that is usually a megaphone for Mr. Trump’s messages, shunted aside the speech. Instead, the hosts opened the three-hour show by highlighting the war in Iran, the wildfire smoke coming from Canada, flooding in Texas and the latest on the cyclospora parasite.

In a later segment about a speech Secretary of State Marco Rubio gave about far-left terrorism, Fox News’s Charlie Hurt credited Mr. Trump for raising concerns about China: “It’s not until President Trump came along and said, ‘Look, this is a real issue. It’s at our shores and we need to confront it.’” But he did not directly address election security.

Other right-leaning outlets took a similar approach. By midday Friday, for instance, Newsmax’s website was highlighting the clash over the Strait of Hormuz, pushing Mr. Trump’s speech below that conflict and the political fight in South Carolina to succeed the deceased Senator Lindsey Graham.

The reaction seemed to reflect disagreements among Republicans over whether Mr. Trump’s almost single-minded focus on his elections bill might help or harm their chances in November’s midterm elections.

“Re-litigating 2020 is a mistake,” said Representative Don Bacon, a retiring moderate Republican from Nebraska. “It’s a losing issue, and most people that are reasonable and aren’t so clouded by partisanship know it’s baloney.”

The lack of enthusiasm highlighted just how entrenched the political debate over Mr. Trump’s claims about election integrity have become, and how unwilling a Republican-controlled Congress has been to heed his command to act on them.

Mr. Trump has made baseless assertions of election fraud for years, largely pointing to the same set of concerns. But with no new evidence to support his claims, lawmakers have had little incentive to change their positions. That continued to be true after a speech that included a slew of claims, some exaggerated or distorted, about potential vulnerabilities to foreign election meddling that the SAVE America Act would not address.

Even Mr. Trump’s staunchest right-wing supporters, who have spent years sifting through documents and interrogating debunked claims about stolen elections, acknowledged that Mr. Trump’s speech on Thursday pulled from familiar refrains.

“Is this new information?” questioned Laura Loomer, a far-right activist and sometime adviser to Mr. Trump, in a post on X about 10 minutes into the president’s speech. (She then used Mr. Trump’s remarks to retread familiar territory, particularly warnings about China’s influence that she has been issuing for years.)

Democrats argued that Mr. Trump offered little beyond the usual litany of grievances about his 2020 election loss. Senator Chuck Schumer, Democrat of New York and the minority leader, had a rapid response operation ready to rebut Mr. Trump’s claims. Lawmakers spread aligned talking points across television appearances and social media posts.

The Republican reaction to the speech was marked by little such coordination. Past prime-time addresses by Mr. Trump have been accompanied by a planned effort in which his fellow G.O.P. lawmakers demonstrate their unity with the president, but such an effort appeared largely absent on Thursday night or Friday.

Not everyone kept silent.

Senator Rick Scott, the Florida Republican who has been one of the most forceful advocates for Mr. Trump’s election measure, called for the Senate to stay in session and use every tool at its disposal to pass it. Representative Tim Burchett, a far-right Tennessee Republican known for his folksy language, thanked the president and urged his colleagues to “pass the dadgum” bill.

But by and large, the vulnerable lawmakers and battleground candidates who are crucial to Republicans’ efforts to hold onto the majority focused elsewhere.

A spokesman for Representative Tom Barrett, a Michigan Republican who is in the middle of a tough re-election fight, said that he did not watch the speech. John Braun, a Republican candidate running in a Washington battleground district against Representative Marie Gluesenkamp Perez, said he “did not see the speech” and “can’t make a comment.”

Some Republicans who did watch came away questioning the president’s choice of focus as his party faces an uphill fight to hold its congressional majorities in November.

“We have 109 days until the midterm elections, and I don’t understand talking about what happened six years ago in light of these upcoming elections,” Senator John Cornyn, the Texas Republican who lost his re-election bid in a primary to a Trump-backed challenger, said on Friday at the Aspen Security Forum. He added: “We ought to be talking about things looking forward that our constituents are most concerned about.”

Ms. Murkowski, who is not up for re-election this year, said that she worried that Mr. Trump’s efforts could shake confidence in the elections and drive down Republican turnout.

“I don’t think it was helpful to us in terms of incentivizing people to get out and participate,” she said.

Senator John Thune, Republican of South Dakota and the majority leader, has faced relentless attacks from right-wing Republicans over his failure to push through the SAVE America Act. As of Friday afternoon, he had not issued a statement on Mr. Trump’s remarks, but he acknowledged before the speech that there was little that could force the president’s elections bill from its logjam.

“If I thought there was a path to getting a positive result or a positive outcome, I’d be willing to do just about anything, because I think we are all here in favor of the SAVE America Act,” Mr. Thune said, adding that it was “very unlikely that the Senate Republicans are going to vote to get rid of the legislative filibuster.”

A version of the bill Mr. Trump wants has stalled in the Senate, where it cannot muster the 60 votes needed to overcome a filibuster in the face of unified opposition from Democrats. Several Republicans have said that there is not enough support to weaken the filibuster just to pass the voting bill. Some Republican senators have said that they oppose the legislation outright.

House Republicans intend to include it in a special budget bill they hope to begin pushing through next week, but the party is divided on that measure. It also is not clear the voting restrictions can be included in the legislation, which must comply with strict rules requiring all of its provisions to have a direct impact on federal spending or revenue.

Stuart A. Thompson and Tim Balk contributed reporting.