Meanwhile Marco Rubio dropped out of the Republican race after losing in his home state of Florida to Mr Trump.
The billionaire is also set to win in Illinois and North Carolina.
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"We have to bring our party together," Mr Trump said in Palm Beach, Florida, as the results came in on Tuesday evening.
He was hoping the votes would give him a clear path to becoming the Republican choice for the White House.
However, John Kasich secured his first win in the Republican contest in the state of Ohio where he is governor.
In his victory speech, Mr Kasich said he wanted to create a "climate of opportunity" for future generations and praised the campaign of Marco Rubio.
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