David Jackson

USA TODAY

Donald Trump swept to primary victories Tuesday in Michigan and Mississippi, putting even more pressure on rivals who are trying to stop his drive for the Republican presidential nomination.

Trump also won the Republican caucuses in Hawaii.

Texas Sen. Ted Cruz did defeat Trump in the Idaho primary.

"There's only one person (who) did well tonight: Donald Trump," the New York billionaire told supporters who gathered at his golf club in Jupiter, Fla.

Claiming victories despite a week of "horrible lies" told by opponents and "special interests" trying to block him, Trump said "it shows you how brilliant the public is."

Cruz narrowly edged Ohio Gov. John Kasich in the battle for second place in Michigan, the biggest prize of the night, while Cruz ran a distant second to Trump in Mississippi.

In his victory remarks, Trump said he would work for Republican Party unity, including support from GOP lawmakers who now oppose him. Trump cited the record turnouts he has helped generate in many Republican contests to date, including Democrats willing to cross party lines.

"I hope the Republicans can embrace it," Trump said.

Kasich told supporters in Columbus that he is already looking toward next week's big primary in his home state: "We are going to win the state of Ohio and it will be a whole new ballgame."

Florida Sen. Marco Rubio, who finished a distant fourth place in both Michigan and Mississippi, spoke to supporters in Florida before any returns were reported and vowed to press on. The Florida senator faces a must-win of his own next week in his home-state primary.

After claiming his wins in Michigan and Mississippi, Trump fired back at one critic in particular: Mitt Romney, the 2012 Republican nominee who said last week he could never vote for the New York billionaire. Trump disputed Romney's claims that he has had many failed businesses by displaying examples of Trump steaks, Trump water, Trump wine and Trump magazine at his news conference.

Earlier Tuesday, Trump dealt with new questions about the now-defunct Trump University and his recent habit of asking supporters to raise their hand and pledge their votes to him — gestures that some critics have likened to Nazi or fascist salutes.

"I think it's ridiculous," Trump told NBC's Today show, saying people are just having a good time. "I think that's a big, big stretch."

Additionally, the Better Business Bureau issued a statement disputing many of the New York billionaire's claims about a former business venture known as Trump University, currently the subject of fraud lawsuits.

"During the period when Trump University appeared to be active in the marketplace, BBB received multiple customer complaints about this business," the bureau said.

Complaints are automatically rolled off after three years, meaning that, over time as the school gradually ceased activity, Trump University's rating rose from a D-minus to an A-plus — the rating Trump has repeatedly cited during discussions of the episode.

"I'll win the lawsuit," Trump told reporters after the wins in Michigan and Mississippi.

Fact check: Trump University�s D- rating

The GOP front-runner also picked up a prominent endorsement ahead of Tuesday's primary: former New York City mayor Rudy Giuliani.

The run-up to Tuesday's delegate contests also included more in-fighting among Trump's opponents.

Rubio and his aides accused the Cruz campaign of spreading false rumors that the Florida senator is being urged to exit the race ahead of next week's Florida primary, where polls have Rubio trailing Trump. The allegations are similar to those that Ben Carson raised against Cruz on the night of the Feb. 1 Iowa caucuses.

The news report cited by Cruz backers and others is "a lie," Rubio told Fox News Radio, and "it looks like Ted Cruz’s campaign is putting out emails in places like Hawaii, telling people about it and you saw that with Ben Carson earlier.”

Rubio aide accuses Cruz camp of �disgusting tactics�

Cruz's aides said that volunteers in Hawaii sent out unauthorized messages about Rubio's prospects.

For his part, Cruz said he looks to maintain momentum he had on Saturday when he "clobbered" Trump in Kansas and Maine.

"The last election day — Super Saturday — was a very bad day for Donald," Cruz, a Texas senator, told reporters.

Meanwhile, Rubio and Kasich now face must-win primaries on March 15 in their home states to keep their campaigns afloat.

Those two candidates and Cruz argue that they are the best-positioned to defeat both Trump in the Republican race and likely Democratic nominee Hillary Clinton in the fall general election.

Rubio, addressing supporters Tuesday night, touted his credentials as a general election candidate, saying he can unite the party and defeat the Democrats in the fall, things he said Trump cannot do because too many Republicans find him to be an unacceptable nominee.

Saying that "I've been an underdog my whole life," Rubio told the crowd: "I believe with all my heart that the winner of the Florida primary next Tuesday will be the nominee of the Republican Party."

Kasich predicted he would start picking up more delegates as the race turned toward Ohio and other Midwest states. "Now the home-court advantage is coming North," he said. "And next week, we're going to win the great state of Ohio."

Trump also predicted success in Ohio and Florida, telling supporters that "I think we're going to clean the slate" and "I like to close things out."

Contributing: Chrissie Thompson, The Cincinnati Enquirer

Elections 2016 | USA TODAY Network