It is a significant victory for the former president and puts him in a commanding position to become the party's presidential candidate in November.
His last remaining rival in the race, Nikki Haley, has vowed to fight on and said "this race is far from over".
Speaking at a campaign stop earlier, front-runner Trump told reporters "I don't care" if Haley stays in the race, adding "let her do whatever she wants".
Joe Biden is projected to win the Democratic primary, but because of a fight within the party those results will not be recognised.
