The Senate bowed to political reality at 4:36 a.m. on Saturday, adjourning and allowing bleary-eyed senators to leave Washington without any concrete steps toward passing the voting restriction bill that President Trump has demanded for months.

Republican lawmakers had insisted they would work tirelessly to enact election changes that Mr. Trump has suggested the party needs in place to win the midterm elections. Instead, they departed with little to show for their efforts, other than a performative pre-dawn procedural vote on a watered-down version of the bill that they knew they could not pass.

Even the staunchest advocates of the election measure eventually conceded what had been clear all along: There was simply not enough Republican support to push through the strict voter identification legislation. With the party’s majority at risk, senators were eager to abandon the futile effort so they could get home to their states to campaign for re-election.

“We are not in a position right now where we have the wherewithal to pass it right now,” Senator Mike Lee, Republican of Utah and one of the most forceful proponents of the legislation, said on Friday night.

Senator John Thune, Republican of South Dakota and the majority leader, had been making that case for months, alluding in increasingly blunt terms to “math” and “reality” that he suggested the president simply did not appreciate as Mr. Trump angrily demanded the Senate find a way to pass the elections bill.

The president’s proposed measure would require Americans to show proof of citizenship to register to vote, impose a nationwide voter identification requirement and severely limit voting by mail. That legislation, which the president has called the SAVE America Act, has neither the majority it would need to pass nor the 60 votes needed to overcome a Democratic filibuster.

Still, Mr. Trump commanded the Senate to cancel its planned five-week recess if it did not pass the measure. Mr. Lee had been one of his loudest allies, threatening to block the Senate from adjourning or force his colleagues to go on the record with a politically unappealing vote in favor of leaving Washington for the summer rather than staying at work at the Capitol.

But in the end, Mr. Lee did neither. Though he forced the Senate to vote to adjourn, he accepted a voice vote that did not require senators to formally register a position.

In a late push to show progress on the elections measure, Republican leaders had rushed to draft a $150 billion budget blueprint to fund the war with Iran and the implementation of new election restrictions, and discussed bringing it to a vote. That plan would pave the way for filibuster-proof legislation known as a reconciliation bill to bypass Democratic opposition.

But it was clear that the measure would have failed overnight, lacking enough support among Republicans to move ahead.

Senator Ron Johnson, Republican of Wisconsin and the chairman of the Budget Committee, said that lawmakers would postpone the effort until they return in September. He and other senators called Mr. Trump to tell him of the plan, which they said he supported.

“We’re in total agreement,” Mr. Johnson said on Friday night. “We were just kind of like, ‘Here’s the reality.’ He would like a better reconciliation bill that will take more time, so he was completely supportive of it.”

But the budget reconciliation process is cumbersome, and Republicans are running out of time to pass their measure before Election Day. The Senate has just three scheduled weeks in Washington before the midterm elections, in which control of Congress is at stake.

Senators who are up for re-election were particularly eager to return home so they can focus on making their cases to voters. Senator Lisa Murkowski pointed to several competitive races, including that of Senator Dan Sullivan, her fellow Alaska Republican, as she made the case for adjourning.

“The most important thing for Senator Sullivan as well as my friend Susan Collins and Jon Husted and Pete Ricketts is for them to be able to go home and be in their states and go to work again,” Ms. Murkowski said, referring to Republican senators in Maine, Ohio and Nebraska who must face voters in November.

Before they left, Republicans held a test vote meant to put Democrats on the defensive on the voter ID piece of the elections bill, seizing upon an idea that polls have shown is broadly popular. They tried to advance a bill sponsored by Mr. Husted, who is facing a difficult re-election race, that would have required voters nationwide to show certain forms of photo identification in federal elections.

Ultimately, the effort failed on a party-line vote, with all Democrats voting against it.

Though some Democrats have said they support federal voter ID requirements, they have argued that Mr. Husted’s measure would too strictly limit what forms of identification were acceptable, potentially blocking eligible voters who lacked the proper documentation from being able to cast ballots.

Catie Edmondson and Carl Hulse contributed reporting.