Republican Donald Trump has vowed to hit rival Democrat Hillary Clinton harder in the next US presidential debate after she put him on the defensive by accusing him of being racist, sexist and a tax dodger during their first matchup.

Key points: Trump says questions were "very unfair"

Trump says questions were "very unfair" Clinton says Trump made claims "that were demonstrably untrue"

Clinton says Trump made claims "that were demonstrably untrue" Ratings show 80.9 million Americans watched debate

Mr Trump, making his first run for public office, praised himself for not attacking Mrs Clinton about the marital infidelity of her husband, former president Bill Clinton, during the debate at Hofstra University.

But he said in a morning interview with Fox News he may take up the attack line going forward.

"I may hit her harder in certain ways," Mr Trump said.

He added that when Mrs Clinton criticised him for his treatment of women, he resisted.

"I was going to hit her with her husband's women. And I decided I shouldn't do it because her daughter was in the room," he said.

Mrs Clinton brushed off Mr Trump's promise, saying: "He can run his campaign however he chooses."

Mr Trump himself had a high-profile affair with Marla Maples, the woman who would be his second wife while he was still married to his first wife, Ivana Trump.

He eventually divorced Ms Maples and married his third and current wife, Melania Trump.

Mrs Clinton went after Mr Trump again the day after her forceful performance in the first of three scheduled presidential debates ahead of the November 8 election.

The New York real estate mogul, she said, "was making charges and claims that were demonstrably untrue, offering opinions that I think a lot of people would find offensive and off-putting".

Mr Trump sought to deflect criticism of his debate performance, saying debate moderator Lester Holt of NBC News asked him "very unfair questions" and that he was given a "very bad" microphone.

Mrs Clinton, speaking to reporters on her campaign plane, said: "Anyone who complains about the microphone is not having a good night."

Mr Trump said Holt failed to press Mrs Clinton on "her scandals", citing the controversies over her use of a private email server as secretary of state and her handling of a 2012 attack on the US consulate in Benghazi.

Record ratings as more than 80 million tune in

Donald Trump and Hillary Clinton meet in the first debate first presidential debate. ( Reuters: Rick Wilking )

The debate was watched by 80.9 million Americans, setting a new viewer record for such events, CNN said.

The figure includes the approximately 45 million who watched the 90-minute debate on the four main TV networks, as well as millions more who watched cable channels, CNN, MSNBC, Fox News and other smaller outlets, CNN said.

The figures, which are not final, beat the 80.6 million record set by the Jimmy Carter-Ronald Reagan TV presidential debate in 1980 but are below the 100 million viewers some analysts had predicted ahead of the face-off.

The day after the debate Mrs Clinton highlighted what she described as Mr Trump's downbeat message.

"He talks down America every chance he gets. He calls us names. He calls us a Third World country. He talks in such dire and dark terms. That's not who America is," Mrs Clinton said.

"The real point is about temperament and fitness and qualification to hold the most important, hardest job in the world."

Mrs Clinton was under pressure to perform well on Monday night after a recent bout with pneumonia and an erosion in recent weeks in her lead over Mr Trump in opinion polls.

Mr Trump, a former reality TV star who eschewed a lot of debate practice, was assertive and focused early on, interrupting Mrs Clinton repeatedly.

As the night wore on, he became testy and less disciplined in front of the crowd.

A CNN/ORC snap poll said 62 per cent of respondents felt Mrs Clinton won and 27 per cent believed Mr Trump was the winner.

In signs that investors awarded the debate to Mrs Clinton, Asian shares recovered after an early bout of nerves while the Mexican peso jumped.

Mrs Clinton's chances in the election also improved in online betting markets.

Reuters