The comments capped a tumultuous day on the campaign trail in which a demonstrator rushed a stage where Trump was speaking.
Trump also accused Bernie Sanders' supporters of sowing unrest at his events and the GOP front-runner refused to back down from his rhetoric that some have cited as the cause of heightened tension at his rallies.
"I'm going to ask that you arrest them," Trump said to the police. "I'll file whatever charges you want. If they want to do this ... we're going to go strongly for your arrests."
Trump said arresting protesters would "ruin the rest of their lives" by giving them a "big arrest mark."
"Once that starts happening, we're not going to have any more protesters, folks," Trump said.
As dozens of protesters disrupted Trump's rally inside a Kansas City theater, crowds of demonstrators gathered outside.
Lining the sidewalks and four corners of the nearest intersection to the theater, the protesters voiced their opposition to Trump's policies and rhetoric, which many characterized as racist, xenophobic and fear-mongering.
Police used pepper spray at least twice Saturday night, with the Kansas City Police Department tweeting that "we had to use pepper spray 2 times outside Trump rally and arrested two people who refused to follow law."
As Trump supporters left the venue, protesters shouted and cursed at them -- and any passersby coming from the direction of the theater -- calling them "f***ing racists."
Alicia Valeanzela, who was shouting those words at people she believed to be Trump supporters, said she believed anyone supporting Trump supports a racist, xenophobic ideology.
"He's a f***ing bigot. He's an a**hole," she said.
"It's not right I'm not gonna let somebody ruin our country like that," said the 21-year-old native of Venezuela. "People need to know that they cannot vote for Trump, and Trump cannot become our President."
Among those targeted by the protesters' cries were the Toates family: father Phillip, his wife and their three children, including a 10-year-old son.
"How do they know I even attended the rally? They say Trump's all about hate, but we have not been about hate and that's the way we got treated when we came out," said Phillip Toates, who said he is leaning toward supporting Trump but is still undecided.
"It's not the way I expected to be treated coming out of a rally," he said.
'Our communist friend'
Trump began the day by blaming supporters of Sanders, the Vermont senator seeking the Democratic presidential nomination, of disrupting his Chicago event, where many in the audience chanted the senator's name after the rally was called off.
"Some represented Bernie, our communist friend," Trump said in Dayton, Ohio, his first campaign appearance since the Chicago event was postponed.
Later in the day, Trump said protesters at his Cleveland event are "Bernie's crowd."
"You know Bernie was saying Mr. Trump should speak to his crowd," Trump said. "You know where they come from? Bernie's crowd. They're Bernie's crowd."
And when a protester momentarily disrupted Trump's rally, the GOP front-runner again said the demonstrator was a "Bernie person."
"Get your people in line, Bernie," Trump said.
Sanders, however, pushed back against Trump, calling on him to denounce violence at his rallies and labeling him a "pathological liar."
'I don't think our supporters are inciting'
The progressive group MoveOn.org, which has endorsed Sanders
, said in a statement Saturday that it helped students print signs for the protests at the Chicago rally and recruit members to attend the "student-led protest."
"I don't think our supporters are inciting. What our supporters are doing is responding to a candidate who has, in fact, in many ways, encouraged violence," Sanders said at a news conference in Chicago. "When he talks about ... 'I wish we were in the old days when you could punch somebody in the head.' What do you think that says to his supporters?"
Sanders also referred to an incident this week in which a black protester was sucker-punched by a Trump supporter as he was being led out of a rally.
"So the issue now is Donald Trump has got to be loud and clear and tell his supporters that violence at rallies is not what America is about and to end it," Sanders said.
In a statement issued later Saturday, Sanders added: "As is the case virtually every day, Donald Trump is showing the American people that he is a pathological liar."
4 people arrested
Clashes broke out Friday night between protesters and Trump supporters after the campaign announced the rally would be canceled more than 30 minutes after it was scheduled to start. Hundreds of protesters had packed into the University of Illinois at Chicago venue for the rally, prompting the campaign to call off the event.
Four people have been charged following the event in Chicago, with charges ranging from aggravated battery to resisting arrest, Chicago Police Department said in a statement.
The protests and fights in Chicago were the latest in a string of increasingly heated and at-times violent confrontations breaking out at rallies for the front-runner in the Republican presidential race. And they come as Trump has repeatedly suggested protesters should face more violent repercussions for disrupting his rallies.
"We're all together and we want to get along with everybody, but when they have organized, professionally staged wise guys, we've got to fight back, we've got to fight back," Trump said Saturday in Dayton.
As he did the previous night in a round of phoned-in TV interviews, Trump didn't walk back any of his rhetoric Saturday. He again claimed that neither the tone of his campaign nor his supporters were to blame for any violence at his rallies.
"They want me to tell my people please be nice, be nice. My people are nice," Trump said Saturday.
"They were taunted, they were harassed by these other people."
Saturday night, Republican National Committee chairman Reince Priebus issued a statement
in which he expressed his hope that "all Americans exercise their First Amendment rights" but do so "in a manner that is respectful of our fellow Americans."
"As we go further along in this election cycle, I ask those on my side of the political spectrum and those on the left to engage actively in the political process in a manner that continues to make our democracy the envy of the rest of the world," Priebus said. "Leaders and activists in both parties bear a responsibility to ensure that the discourse we engage in promotes the best of America."
Man tries to rush stage
While Trump was speaking in Dayton, a man jumped a barricade behind the stage in an apparent effort to confront the GOP front-runner, but was unsuccessful. He was quickly blocked by Secret Service officers, and members of Trump's security detail rushed to cover him. After a few moments, Trump, who was visibly startled, gave a thumbs-up and thanked the cheering crowd for warning him.
"I was ready for him," Trump said. "But it's much easier if the cops do it."
Montgomery County Sheriff Phil Plummer identified the protester as Thomas DiMassimo, 22, of Fairborn, Ohio. He was charged with disorderly conduct and inducing panic, according to authorities, and was being held at Montgomery County Jail in Dayton early Saturday afternoon.
A spokesperson for the jail said DiMassimo has a court appearance scheduled for Monday.
In a statement, DiMassimo called for peaceful protests.
"We must nonviolently shut down every Trump rally," he said. "We must stop this violent, foolish man from taking this country and destroying it."
Trump spokeswoman Hope Hicks issued a brief statement on the incident.
"A man attempted to breach the secure buffer and was removed rapidly and professionally," she said. "All further inquiries should be directed to the (Secret Service)."
Dispute over decision to cancel rally
Early Saturday afternoon, Trump's campaign issued a statement about the Chicago rally, saying police, as well as Secret Service and private security firms, "were consulted and totally involved" in the decision to cancel the event.
"We have received great credit from everyone for canceling this event. Nobody was injured and crowds disbanded quickly and peacefully. It has been termed 'really good management and leadership under great pressure!'" the statement read. "It would have been easier for Mr. Trump to have spoken, but he decided, in the interest of everyone's safety, to postpone the event."
Trump campaign manager Corey Lewandowski added, "We don't make decisions in a vacuum. We coordinate with all the appropriate law enforcement."
But Anthony Guglielmi, a Chicago police spokesman, told CNN that Trump's claim that he informed a police commander before making the decision to cancel the rally was "absolutely untrue."
"No one from the campaign contacted the Chicago Police Department and we were not involved at all in canceling the event," Guglielmi told CNN.
"I was literally standing next to the superintendent when we were notified by the university that the event had been canceled. I think a lot of people were surprised, including the Secret Service officer standing next to us who was notified at the same time."
Guglielmi added that police then dispatched an additional 100 police officers to the University of Illinois arena, where the event was being held, to assist with crowd control.
Before Trump's rally in Kansas City, a police supervisor at the scene earlier in the day said the department had not deployed more officers than usual, though there is a large force on hand.
But, said the supervisor, who requested anonymity because he was not authorized to speak on the record, "We are certainly aware of what happened in Chicago and St. Louis," where more than 30 people were arrested Friday afternoon at a Trump rally.
2016 hopefuls continue criticism
The other 2016 presidential candidates continued their attacks on Trump on Saturday over the violence at the Chicago rally.
"If you play with matches, you're going to start a fire you can't control," Hillary Clinton said at a caucus kick-off event at a local YMCA in St. Louis
. "That's not leadership. That's political arson."
Florida Sen. Marco Rubio said he still plans to support the Republican nominee in the general election, but it's "getting harder every day."
"I think we also have to look at the rhetoric coming from the front-runner in the presidential campaign," he told reporters in Largo, Florida. "Someone who's basically encouraged people in the audience to rough up anyone who stands up and says something he doesn't like."
Rubio said if the anger from voters continues, the country will "continue to be ripped apart at the seams."
Texas Sen. Ted Cruz accused Trump of encouraging his supporters to physically assault activists.
"Unlike Donald Trump, I don't ask people in the audience to punch people in the face. I don't ask them to engage in physical violence against protesters," he said in Kansas City. "And I think the way you interact with the citizens expresses what kind of President you'll be."
Ohio Gov. John Kasich -- who has made running a positive campaign a central part of his appeal to voters -- initially declined to criticize Trump on Friday night, saying he couldn't comment on the rallies as he had never been to one. Later, he issued a statement, saying Trump had been sowing "seeds of division" that "finally bore fruit."
But by Saturday morning, Kasich -- who, like Cruz, had declined to blame Trump for violence at his rallies, called a news conference to denounce Trump.
"Donald Trump has created a toxic environment, and a toxic environment has allowed his supporters and those who sometimes seek confrontation to come together in violence," Kasich told reporters in Sharonville, Ohio. "To see Americans slugging themselves at a political rally deeply disturbed me. We're better than that."
CORRECTION: This story has been updated after the Montgomery County Sheriff's office misstated DiMassimo's age.