In the latest and most dramatic sign of the challenge Donald Trump faces in uniting the Republican Party behind his candidacy, House Speaker Paul Ryan said Thursday that he couldn’t yet lend his support to the party’s presumptive presidential nominee.

Mr. Ryan, who will be chairman of the Republican national convention in July in Cleveland, said he hoped to come around to supporting Mr. Trump. But in the meantime he took a remarkable step by giving a high-profile voice to anxieties running through the party about Mr. Trump’s...