Speaking in a subdued tone, and briefly choking up, Mr. Bush did not mention Mr. Trump but said it was essential that the next president be a person who would serve “with honor and decency.”

If the most strenuous test of Mr. Trump’s staying power is still to come, he proved in South Carolina that he would not be easy to stop. Using blunt and at times incendiary language, he found strong support from Republicans without a college degree, those who are angriest about the federal government and those who favor a hard line on illegal immigration, according to surveys of voters leaving the polls.

Mr. Trump’s victory here was not as sweeping as the one in New Hampshire, and there were warning signs as he showed weakness among women and younger voters. But he still won among both independents and Republicans, and among self-described evangelical Christians. He also seemed to have built a coalition that will remain with him through adversity: More than half of voters who made their decisions over a month ago picked Mr. Trump, exit polls showed.

That support held through a tumultuous week in South Carolina, during which Mr. Trump encountered harsh attacks and invited controversy. He faced severe criticism for his past support of abortion rights and for trying to evict an elderly woman from her home to build a casino parking lot in Atlantic City. And he ridiculed President George W. Bush’s handling of terrorism and the Iraq war, criticisms assumed to be ill-considered in any Republican primary, but particularly in a military-rich state that delivered victories to Mr. Bush and his father.