In Georgia, which faces a tight race, officials warn that the vote tally is incomplete.

Representative Chip Roy, a member of the House Freedom Caucus, successfully held off a challenge from Wendy Davis, a Democrat who gained national attention in 2013 for filibustering an anti-abortion bill and later ran for governor.

In the Houston suburbs, where Representative Pete Olson’s retirement led to a competitive race, Sheriff Troy Nehls of Fort Bend County, a Republican, defeated Sri Preston Kulkarni, who narrowly lost to Mr. Olson in 2018.

It was still possible for two Democrats to pick up seats in the state early Wednesday: Gina Ortiz Jones, a former Air Force intelligence officer, is making a second run for Texas’s 23rd Congressional District, and Candace Valenzuela, a Democratic school board member who would be the first Afro-Latina member of Congress, is running in the 24th Congressional District.

Because absentee ballots are manually opened and placed into a scanner before tabulating, they require extra time. In some cases, ballots kicked out by scanners as ambiguous must be reviewed by special adjudication panels. Richard Barron, the director of elections for Fulton County, said this week that of a batch of 40,000 ballots processed by his office, approximately 700 required review.

Delays in counting ballots were being reported in other metro counties as well. Some ballots placed in drop boxes on Tuesday could not be collected until after polls closed at 7 p.m.

Secretary of State Brad Raffensperger had said that he hopes results from most races in the state will be available Wednesday.

“We will win this,” he continued. “As far as I am concerned we already have won it.” He had not, in fact, won the battleground states he claimed as victories, like North Carolina, Pennsylvania and Michigan. Many states were still counting votes and had not reported any vote totals.

His remarks were an escalation of his monthslong effort to cast doubt on the legitimacy of the race and previewed what is expected to become a protracted legal battle. Before Trump’s false claims, his campaign was already fund-raising off the uncertain outcome. A 12:03 a.m. plea asked for money to help “protect the integrity of this Election.”

They also stood in contrast with remarks made earlier in the evening by former Vice President Joseph R. Biden Jr., who urged his supporters to have patience until all the votes were counted.

“It’s not my place or Donald Trump’s place to declare the winner of this election,” Mr. Biden tweeted. “It’s the voters’ place.”

An early call of Arizona for Mr. Biden by Fox News was another check on any legitimate claim that Mr. Trump had won, or was even the likely victor in the race. And that call appeared to have enraged him. He told supporters that he had been watching Arnon Mishkin, the leader of the network’s decision desk, who appeared just after 12:30 a.m. and insisted Mr. Trump could not win the state.

“We have a lot of life in that and somebody declared that it was a victory,” Mr. Trump said. “Maybe it will be. I mean that is possible. But certainly there were a lot of votes out there that we could get because we’re now just coming into what they call Trump territory.”

“I’m confident that when the votes are counted, we’re going to be successful in this mission,” he said.

Mr. Kelly thanked his wife, Gabrielle Giffords, the former Democratic congresswoman who survived a 2011 assassination attempt. He said she taught him about public policy and public service, and also “to never ever give up.”

Ms. McSally launched her political career in the wake of the shooting of Ms. Giffords, running three times for her old congressional seat before securing a win.

Mr. Kelly highlighted his support for the young undocumented immigrants known as Dreamers, as well as his pledges to secure the border, part of the bipartisan message that he employed throughout the campaign.

“Some of you watching tonight did not vote for me,” he said. “That’s OK. I’m going to be your senator, too.” He invoked the name of John McCain, whose seat he expects to fill. If his victory is confirmed, Mr. Kelly will be up for re-election in 2022, when the term of Mr. McCain, who died in 2018, would have ended.

The McSally campaign did not concede defeat, and instead issued a statement that charged Fox News as “irresponsible” for calling the race. The McSally campaign’s statement echoed complaints from the Trump campaign, which had publicly criticized Fox for declaring Joseph R. Biden Jr. the winner of the state in the presidential race.

“There are more than 1 million votes to be counted with no Election Day votes yet reported,” Ms. McSally’s campaign spokeswoman Caroline Anderegg said.

Ms. McSally lost a 2018 U.S. Senate race against Kyrsten Sinema before being appointed to Mr. McCain’s seat after his death. Her defeat would be the first time in modern history that a candidate has lost both of a state’s Senate seats to the opposing party.

President Trump, who has strong support in South Carolina and handily won the state on Tuesday, had repeatedly praised Ms. Mace on Twitter over the course of the race. Both candidates had campaigned with their respective party’s Senate nominee, with Ms. Mace appearing with Republican incumbent Senator Lindsey Graham, who won his race, on the weekend before Election Day.

“Nancy Mace is Nancy Pelosi’s worst nightmare,” Mr. Graham said at a rally over the weekend, calling Mr. Cunningham “nice” but an aberrant choice for South Carolinians.

“She’s not going to talk one way and vote another,” Mr. Graham said of Ms. Mace. “She’s going to vote like you would want her to, the conservative values that made the low country of South Carolina someplace special.”

The second district here is in large part a mishmash of cul-de-sacs and cookie-cutter housing developments that blur into each other. It also includes a population that has become increasingly less white, notably with more Latinos moving into South Omaha.

While Keep America Great signs dot the rural parts of the state, signs for Mr. Biden are propped up across the wide lawns in the tony, historic district of Dundee-Happy Hollow and in front of homes and business in the largely Black neighborhood of North Omaha, the birthplace of Malcolm X.

In Omaha, like in most of Nebraska, cases of coronavirus are soaring but outside of hospital wards and facilities for senior citizens under lockdown, life seems relatively normal. Schools are open, restaurants and bars are less crowded but allow some indoor patrons, high school football playoffs took place last week, a major milestone here in the transition to the holiday season.

On Tuesday, voting in Omaha proceeded more or less as normal, with more hand sanitizer.

Representative Don Bacon, a Republican, fresh from being chided by Mr. Trump at a rally in Omaha, as well as in a phone call, to deliver the district for him, paused before he voted at a polling site in a suburban church to consider what it would take for Mr. Trump to win.

“I always thought if he could make this about results versus the Twitter and the name calling — I don’t think this district responds well to that,” he said.

Many people who were voting for Mr. Trump cited the economy as the main reason they supported him. They blamed the recent downturn on the virus and said Mr. Trump could not be held responsible for something that they said China should take the blame for.

“He didn’t have nothing to do with it,” said Chuck Kopystynsky, a 70-year-old former Marine who was voting at a Honda dealership that had been converted to a polling place because the usual spot was an assisted-living facility housing seniors vulnerable to the virus.

His wife, Vikki Kopystynsky, said she liked the president’s stance on immigration. She resents the idea that Democrats want to give free college education to “Dreamers,” who were brought to the country illegally as children.

“My daughter couldn’t go to college because we couldn’t afford it,” said Ms. Kopystynsky, who is white.

Allison Lindgren, a 26-year-old stockbroker from Millard in southwest Omaha, said she had never even stopped to consider that Mr. Biden could lose. If that happens, she said, “The first thing we’re all going to do is go to the internet,” and post about feelings of outrage. “I don’t think the country can stand another four years of him. I don’t want him representing us to the world anymore.”

Facebook did not restrict users from sharing or commenting on the post. It was the first time Facebook had used such a label, part of the company’s plan to add context to posts about the election.

A White House spokesman did not immediately respond to a request for comment.

In a tweet sent about 10 minutes after Mr. Trump’s post, the Democratic presidential candidate, Joseph R. Biden Jr., said: “It’s not my place or Donald Trump’s place to declare the winner of this election. It’s the voters’ place.”

Twitter began fact-checking and labeling Mr. Trump’s misleading tweets in May. Mr. Trump responded with an executive order designed to strip legal protections from Twitter and other social media companies. Facebook has also increased its initiatives around protecting the election.

Under the measure, the state minimum wage would rise from its current hourly rate of $8.56 to $10 in September, and then increase by $1 every September through 2026. After that, annual increases would be tied to inflation.

A study by the Florida Policy Institute, a think tank backing the increase, found that the higher wage would directly benefit 2.5 million workers in the state.

A number of studies have found that moderate increases in the minimum wage have not led to significant job losses. But economists caution that the effects on employment depend on the size of the increase relative to a city or state’s wage scale.

That could make a $15 minimum wage more costly in a state like Florida, where wages tend to be substantially lower than wages in other states that have enacted a $15 minimum wage.

The victory of Ms. Bice, 46, returned the Fifth District to Republicans, who had held it for more than 40 years before Ms. Horn’s stunning upset in 2018. Republicans had singled out Ms. Horn’s seat, which represents Oklahoma City, as a likely opportunity to counter Democratic gains anticipated in other parts of the country.

Seen as a rising star in the State Legislature, Ms. Bice will bolster the ranks of Republican women in the House, whose numbers have shrunk even as Democrats have made history with the racial and gender diversity of their members. She overcame a tough primary and runoff race by framing herself as more moderate than her opponents, though she later highlighted her conservative credentials and an endorsement from the National Rifle Association.

“That’s not true,” Mr. Mishkin said. “I’m sorry, the president is not going to be able to take over and win enough votes.”

He added: “We’re not wrong in this particular case.”

With almost 80 percent of the Arizona vote counted late Tuesday, Mr. Biden was leading Mr. Trump by more than six percentage points.

Mr. Mishkin faced skepticism from some pro-Trump Fox News colleagues, as well. The conservative pundit Katie Pavlich, a native Arizonan, told viewers she was doubtful about the network’s call, and the host Tucker Carlson said on air that Trump officials were deeply skeptical that Mr. Biden had won the state.

In 2012, Mr. Mishkin appeared on-air to explain to Fox News viewers why he had called Ohio for Barack Obama, a projection that one of the network’s analysts, Karl Rove, had doubted. Mr. Obama ultimately won the state.

Third-party candidates received an unusually large share of the vote — 36 percent — a reflection of Puerto Ricans’ growing dissatisfaction with the two dominant political parties.

A victory by Mr. Pierluisi would keep the governorship in the hands of the New Progressive Party, which supports statehood for Puerto Rico. But unlike his predecessor, Wanda Vázquez, who is a Republican, Mr. Pierluisi is a Democrat. Mr. Pierluisi served briefly as governor last year, after the resignation of former Gov. Ricardo A. Rosselló.

Political parties on the island do not match up with those on the mainland. Mr. Delgado’s Popular Party supports keeping Puerto Rico’s current status as a U.S. territory.

Puerto Ricans also voted on Tuesday on a nonbinding referendum that asked, “Should Puerto Rico be admitted immediately into the union as a state?”

Statehood was leading with about 52 percent of the vote.

The previous threshold was 50,000 votes in a race for governor.

The Working Families Party has sparred with Mr. Cuomo in recent years. The new rules posed such a risk to the party’s electoral influence that it directed significant financial resources away from competitive races and toward preserving its ballot line.

“Working people and Black and brown communities are historically challenged when we grow in power — and the threat to our ballot line is no different,” Sochie Nnaemeka, the Working Families Party’s state director for New York, said in a statement Tuesday night. “What’s clear, though, is our campaign to protect the ballot line has only resulted in a stronger, more united progressive movement in New York State.”