"Kasich shouldn't be allowed to run. Honestly, Kasich should not be allowed to run," Trump told reporters Sunday while visiting a diner in Milwaukee, adding, "He hurts Trump much more than he hurts (Ted) Cruz."
He pointed out the Ohio governor has only won one contest, the primary in his home state.
"Kasich shouldn't be allowed to continue, and the RNC (Republican National Committee) shouldn't allow him to continue," Trump said.
The businessman said he brought up the issue during his meeting with Republican National Committee Chairman Reince Priebus last week.
"I said, 'Why is a guy allowed to run? All he's doing is he goes from place to place, and loses and he keeps running,'" Trump said. "He doesn't have to run and take my votes. Because he's taking my votes. He's not taking Cruz's votes."
On Monday, Kasich responded while campaigning in New York.
"Donald Trump said I need to get out of the race because I'm going to take his voters, well, I am going to get his voters, because I know what his voters are worried about," Kasich said. "There's a concern that they will never be rich, and that's where the Trump voters are, and that's where I grew up. I grew up with income insecurity and job insecurity."
Chris Schrimpf, a spokesman for Kasich's campaign, dismissed Trump's call on Sunday.
"Ted Cruz also has no possibility of accumulating enough delegates, and Donald Trump also will not receive a majority of delegates before the convention. Since he thinks it's such a good idea, we look forward to Trump dropping out before the convention," Schrimpf said.
"Trump living up to his own self-declared standard is best for the party since he will lose the White House by a historic margin to Hillary Clinton and also cause Republicans to lose control of the Senate," he added.
It is possible for Cruz to still get enough delegates to capture the nomination, but he would have to win almost all of the remaining ones available, which is not considered a realistic possibility.
RNC Chair Priebus told CNN's Jake Tapper on "State of the Union"
Sunday the party's standard bearer will be one of the three men still running -- and did not call for anyone to get out.
"Our candidate is someone who is running," he said.