Even the massive Republican presidential field isn’t enough to contain Donald Trump, the billionaire businessman who on Tuesday inserted himself into Democrats’ race as well by live-tweeting their first presidential debate.

It’s another indication of just how thoroughly he has come to dominate the battle for the White House and how Republican and Democratic candidates have come to realize that their path to the White House runs through Mr. Trump.

The billionaire is enjoying every minute of it, gloating over his 4.4 million followers on Twitter and millions more on Facebook, whom he regaled with thoughts on what he pronounced — even before it started — a “boring” Democratic debate.

“[Russian President Vladimir] Putin is not feeling too nervous or scared,” Mr. Trump said in a post as the candidates introduced themselves at the top of the CNN broadcast.

He then proceeded to dissect the Democratic field the same way he has analyzed his Republican opponents, saying former Maryland Gov. Martin O’Malley “has very little chance” and that he doubted anyone could see former Rhode Island Gov. Lincoln Chafee as president.

“Sorry, there is no STAR on the stage tonight,” he said, echoing the conclusions of a number of instant analysis from pundits online.

That Republicans are taking on Mr. Trump is no surprise. He has shot to the top of the field and remained there for three months. His every word — with planned policy statements and one-off lines in interviews — have moved the debate.

But Democrats are increasingly finding that they also have to grapple with Mr. Trump.

Eager to cloak herself in the attention that follows Mr. Trump, Hillary Rodham Clinton, the front-runner for the Democratic nomination, joined a rally outside the businessman’s Las Vegas hotel Monday night to show solidarity with labor unions demanding that he offer better pay, hours and working conditions.

“Some people think Mr. Trump is entertaining. But I don’t think it’s entertaining when somebody insults immigrants, insults women. That is just unacceptable behavior,” Mrs. Clinton said.

During Tuesday’s debate Sen. Bernard Sanders took aim at “Donald Trump and his billionaire friends,” vowing to make them pay more in taxes.

And former Maryland Gov. Martin O’Malley called him a “carnival barker,” accusing him of being a xenophobe on immigration.

Kevin Sheridan, a Republican strategist who did countermessaging around Democrats’ events at the Republican National Committee in 2004, called the Trump live-tweeting “fairly brilliant.”

“Donald Trump is a countermessaging dream and nightmare all in one. Trolling the event with the biggest megaphone in America practically puts him on the stage with the candidates as every reporter will be looking to cherry pick his best responses throughout an otherwise snoozer of a debate,” Mr. Sheridan said. “Better for everyone if he sticks to insightful mockery rather than cheap shots, but at this point, who knows? He speaks for himself. Give him credit: The guy can steal a spotlight.”

Mr. Trump managed to steal even more attention from Democrats when NBC announced Tuesday that he would host “Saturday Night Live” on Nov. 7.

That drew a rebuke from Hispanic rights group National Council of La Raza, which accused the network and the program of giving a platform to “hate” from Mr. Trump.

“It is appalling, then, that a show with that history and that role to showcase a man whose campaign has been built on bigotry and demagoguery for the sake of buzz and ratings,” said Janet Murguia, president of the National Council of La Raza.

NBC may have been looking at what Mr. Trump could do for its ratings. He took credit — and analysts said probably rightly so — for boosting viewership of the first two Republican debates. The Fox News affair drew 24 million viewers, and CNN’s debate garnered 23 million.

While widely watched, Mr. Trump is not universally loved. Likely Republican primary voters give him the edge in the nominating contests, but general election voters put him near the bottom of the list of Republican candidates in head-to-head matchups with Democrats.

But love him or hate him, his brash style has infected the campaign.

A questioner at a town hall meeting Tuesday in New Hampshire suggested that perhaps Ohio Gov. John Kasich, whose campaign has foundered, could boost his bid by taking a more Trump-like approach on the campaign trail.

“Yell from the housetop? I got you,” Mr. Kasich said — though he said that wasn’t his style.

“It is just hard work, and you can’t pay attention to all the flash and dash that is out there, and by the way I am not going to say anything to get everybody all stoked up so I can get your vote. Then I would not be true to myself, and I am going to go through this campaign as best I can raising the bar about public service and experience in all that,” he said.

• S.A. Miller contributed to this report.

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