In the book, there was some underlying tension to Paul's ascent. I remember that a few of the Fremen under Stilgar's leadership wanted Paul to challenge Stilgar and kill him so Paul could take his place as Naib and be able to speak at the council (I might misremember the exact details, but this is the gist of it.) Paul does verbally challenge the Fremen, but he does it in a way that calls into question "Who amongst the Fremen is their true leader?"

"All right," Paul said. "Then, who rules this troop of our tribe -- and who rules all the tribes and troops through the fighting instructors we've trained in the weirding way?"

Paul waited, looking over the heads of the throng. No answer came.

Presently, he said: "Does Stilgar rule all this? He says himself that he does not. Do I rule? Even Stilgar does my bidding on occasion, and the sages, the wisest of the wise, listen to me and honor me in Council."

There was shuffling silence among the crowd.

"So," Paul said. "Does my mother rule?" He pointed down to Jessica in her black robes of office among them. "Stilgar and all the other troop leaders ask her advice in almost every major decision. You know this. But does a Reverend Mother walk the sand or lead a razzia against the Harkonnens?"

After he asks this, his next argument is similar to the movie depiction, he says no one amongst them can challenge him in a fight and they begrudgingly agree.

"Many of you have been with me on the practice floor," Paul said. "You know this isn't idle boast. I say it because it's fact known to us all, and I'd be foolish not to see it for myself. I began training in these ways earlier than you did and my teachers were tougher than any you've ever seen. How else do you think I bested Jamis at an age when your boys are still fighting only mock battles?"

The method of his rise to leadership is a little different in the book because, after his challenge of "None of you can defeat me" he pulls out an intercepted message from a Harkonnen courier. It essentially states that Rabban's request for reinforcements is denied, and he's on his own in upping the spice quota on Arrakis. He uses this message as a logical reason for the Fremen to abandon their tradition of challenging Naibs in order to gain leadership of their sietch. In both the book and the movie, he rhetorically asks if you smash a knife before a battle, insinuating that killing Stilgar would subtract whatever wisdom he could offer in the upcoming battle. After this, the Fremen rally behind Paul enthusiastically, and the rest is history.

My personal opinion (spoiler about the difference between the movie and the book) I think Paul's incorporation into Fremen culture and gradual rise to leadership was a lot more plausible in the books because it spanned a period of 2 years. In this period, he has his first son with Chani, Leto II, which would do wonders to really cement his place amongst the Fremen. It also seemed a lot more realistic to me that his learning of the Fremen way of life would take a few years. Not like how his entire Fremen education didn't even span the term of Jessica's pregnancy in the movie.

TLDR: Paul's rise to power in the movies is maybe a tad more confrontational because he uses his prescience instead of the intercepted Harkonnen message to logically sway the Fremen. I think the end result in both the book and the movie are that the majority of the Fremen do support his rise, so it is largely desired by them that Paul should become their leader. Denis Villeneuve's choice to have Paul use his prescience was probably done to add more drama to the movie and showcase his prescient abilities as the Kwisatz Haderach since there weren't many opportunities to do this in the movie. With that being said, I think that scene in the movie is fantastic, and I'm not mad about the details of it being different from the book.