The rift in the Republican Party grew deeper on Sunday and threatened to upset the July convention as Donald J. Trump refused to rule out blocking Paul D. Ryan, the speaker of the House, from serving as the convention’s chairman.

Mr. Trump’s warning was his latest affront to Republicans who have urged him to adopt a more cooperative and unifying tone. And it amounted to an extraordinary escalation in tensions between the party’s presumptive nominee and its highest-ranking officeholder.

In a series of television interviews that aired Sunday, Mr. Trump demonstrated little interest in making peace with party leaders like Mr. Ryan who have called on him to more convincingly lay out his commitment to the issues and ideas that have animated the conservative movement for the last generation.

“I’m going to do what I have to do — I have millions of people that voted for me,” Mr. Trump said on ABC’s “This Week.” “So I have to stay true to my principles also. And I’m a conservative, but don’t forget, this is called the Republican Party. It’s not called the Conservative Party.”