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The New York businessman, unpopular with many in his own party, now has a clear path to the 1,237 delegates needed to claim his party’s crown.
That would mark a stunning victory for a businessman few took seriously when he launched his campaign in 2015.
Meanwhile, Bernie Sanders has defeated Hillary Clinton in Indiana’s Democratic race.
Bernie Sanders trails Hillary Clinton in the all-important delegate count but after this victory he said the contest was still alive.
“Clinton campaign thinks this campaign is over. They’re wrong,” he said.
Ted Cruz’s advisers had targeted Indiana as the Texas senator’s best hope of halting Donald Trump’s march to the nomination.
“We gave it everything we’ve got, but the voters chose another path,” Ted Cruz told supporters in Indiana.
Ted Cruz’s departure means Donald Trump is now the presumptive Republican nominee, with plenty of state contests this month and next to reach the 1,237 delegates required to win.
Donald Trump is the first nominee since Dwight Eisenhower in 1952 to lack any previous experience of elected office.
Ohio Governor John Kasich has vowed to remain in the Republican race, but trails far behind Donald Trump in terms of delegates.
“It is a beautiful thing to watch, and a beautiful thing to behold,” Donald Trump said during a victory speech in Indiana.
“We are going to make America great again.”
Donald Trump praised Ted Cruz as a “tough, smart competitor”, which marked a sharp reversal in tone after a day when the two men slung mud at each other from close quarters.
The verbal attacks reached a new level of intensity when Ted Cruz attacked Donald Trump as a “pathological liar” and “serial philanderer”.
That was provoked by a bizarre claim from Donald Trump that Ted Cruz’s father was linked to one of the most traumatic episodes in US history, the assassination of President John F. Kennedy.
It is now increasingly likely that Donald Trump will face Hillary Clinton in the autumn in the battle to succeed President Barack Obama, who will be leaving the White House after two terms.
However, Republicans have expressed reservations about Donald Trump’s outspoken remarks, which have offended women and Hispanics.
There are also concerns about some of Donald Trump’s policies on immigration and national security, like building a wall on the southern US border paid for by Mexico, a ban on Muslims coming to the US and the killing of the families of terrorists.
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