Donald J. Trump won sweeping victories across the South and in New England on Tuesday, a show of strength in the Republican primaries that underscored the breadth of his appeal and helped him begin to amass a wide delegate advantage despite growing resistance to his candidacy among party leaders.

But Senator Ted Cruz reasserted himself with victories in his home state of Texas and neighboring Oklahoma, earning a reprieve as he fends off questions about his viability and bolstering his case that he is the only alternative capable of overtaking Mr. Trump.

Mr. Trump’s political coalition — with lopsided victories in Massachusetts, Georgia, Alabama and Tennessee, and narrower ones in Arkansas and Virginia — appears to have transcended the regional and ideological divisions that have shaped the Republican Party in recent years. He dominated in moderate, secular-leaning Massachusetts just as easily as he did in the conservative and heavily evangelical Deep South.