Democrats were Wednesday celebrating victory in a US special election being viewed as the last bellwether of the public mood on abortion ahead of November's midterms, as the party seeks to make reproductive rights a key issue in the campaign.

Tuesday's vote in a swing district in upstate New York was to choose a candidate to serve the final months of Democrat Antonio Delgado's term in the House of Representatives, after he quit to become the Empire State's lieutenant governor.

The battle for New York's 19th congressional district comes two months after the Supreme Court sparked nationwide protests by ending the federally guaranteed right to abortion.

Democrat Pat Ryan, who has sought to turn the vote into a referendum on abortion access, was projected to win, according to US media including NBC, ABC and Fox News.

"Choice was on the ballot. Freedom was on the ballot, and tonight choice and freedom won," Ryan tweeted after he was forecast to triumph.

"We voted like our democracy was on the line because it is. We upended everything we thought we knew about politics and did it together."

He has accused Republican rival Marc Molinaro and his party, who are against such access, of being "too extreme on women's rights."

Ryan's victory "sends a clear message that voters are fighting back against Republicans' extreme attacks on abortion rights," Democratic Congressional Campaign Committee Chair Sean Patrick Maloney said in a statement quoted by NBC.

"Republicans can say goodbye to their 'Red Wave' because voters are clearly coming out in force to elect a pro-choice majority to Congress this November."

Molinaro, who had followed his party's main political talking points including on inflation and spiralling living costs, had refused to concede, The New York Times reported.

"Whether it's tonight or it's Nov. 8, we are going to win the 19th Congressional District and give voice to people who are working too damn hard and getting too little in return," Molinaro told his supporters late Tuesday, the paper said.

'Right to choose on the ballot'

President Joe Biden carried the district by fewer than two points in 2020, after Donald Trump took it by about seven points in 2016. Barack Obama won there in 2012.

Both candidates will be back on the ballot in November, but in different districts newly drawn in the latest round of redistricting.

The somewhat anachronistic system means Ryan -- who has yet to be nominated to fight for a seat in the midterms -- was on the ballot Tuesday in two districts for two separate seats for two different congresses.

New York also held several nominating contests -- known as primary elections -- including a vote pitting Democratic committee chairs Jerry Nadler and Carolyn Maloney against one another.

Nadler handily won, taking 55.4 percent to 24.4 percent for Maloney with more than 95 percent of votes counted.

Maloney said she had congratulated Nadler on his win.

"I share his progressive values. I wish him every success," she tweeted.

"We must maintain a Democratic House and increase margins in the Senate to stop the extremists who are working to end our democracy," she added.

In Florida, Democrats picked congressman Charlie Crist as their candidate to challenge Governor Ron DeSantis in November.

"Our fundamental freedoms are literally on the ballot, my friends," Crist said in his victory speech, vowing to end the White House hopes of DeSantis, who is tipped as a likely Republican hopeful in 2024.

"A woman's right to choose on the ballot. Democracy on the ballot. Your rights as minorities are on this ballot," Crist said.

"That's what's at stake in this election, make no mistake about it, because this guy wants to be president of the United States of America and everybody knows it.

"However, when we defeat him on November 8, that show is over."

US voters will decide control of Congress in the midterm elections, with all 435 House seats up for grabs, as well as 35 of the 100 Senate seats and the governor's mansion in 36 out of 50 states.