The businessman, who won at least three more states on Tuesday but does not yet have the needed majority of delegates to secure the nomination, predicted he would collect enough support before the Republican convention this summer.
“There’s going to be a tremendous problem” if the Republican establishment tries to outmanoeuvre him at the convention, Mr Trump said. Many party leaders have been concerned about his comments against Muslims, immigrants and women and the violence at some of his rallies.
Democrat Hillary Clinton, who moved closer to becoming the first woman in US history to win a major party nomination after winning at least four states, took direct aim at Trump.
“Our commander-in-chief has to be able to defend our country, not embarrass it,” Mrs Clinton said in a speech that largely ignored rival Bernie Sanders. “We can’t lose what made America great in the first place.”
With anti-Trump Republicans frantically seeking scenarios to deny the billionaire the party nomination, Trump suggested to morning TV shows that the party establishment already was starting to support him.
Without naming names, Mr Trump said some of the same Republican senators who are publicly critical of him have called privately to say they want to “become involved” in his campaign.
Mr Trump also said Mrs Clinton, the former secretary of state and first lady, would be “a major embarrassment for the country” and added that she “doesn’t have the strength or the stamina to be president”.
The Republican front-runner also said he would skip a debate scheduled for Monday, saying: “I think we’ve had enough debates.”
Mr Trump won in Florida, North Carolina, and Illinois but fell in Ohio to that state’s governor, John Kasich. Clinton triumphed in the Florida, Illinois, Ohio and North Carolina primaries.
Florida Senator Marco Rubio dropped out after losing his home state to Mr Trump.
With more than half the delegates awarded through six weeks of primary voting, Mr Trump’s Florida victory brought his delegate total to 621. Mr Cruz has 396 and Mr Kasich 138. Mr Rubio left the race with 168.