COLUMBUS, Ohio — Since the Republican Party’s 19th-century founding, not one of its nominees has won the White House without carrying Ohio, a most diverse and perennially hard-fought state.

But as Republicans head to Cleveland to nominate Donald J. Trump in two weeks, a convention city chosen with Battleground Ohio much in mind, a vortex of headwinds are rising against Mr. Trump in the state.

The barely concealed disdain of Gov. John R. Kasich, a former rival who has not endorsed the presumptive nominee, echoes through the state’s Republican leadership, whose full engagement in the fall campaign will be needed to turn out voters.

Images of disunity in Cleveland, where delegates are gathering July 18-21 in the shadow of local polls showing a majority of Republicans prefer a different nominee, could make it harder for the party to attract grass-roots activists for the fall campaign.