“I will be the nominee of our party, Chris,” she told Mr. Cuomo. “There is no way I won’t be.”

On Thursday afternoon, Michael Briggs, a spokesman for Mr. Sanders, said in a statement that his candidate’s recent victories in Indiana, West Virginia and Oregon showed that voters there “respectfully disagreed with Secretary Clinton.” He added, “We expect voters in the remaining eight contests also will disagree,” and said that some polls showing Mr. Sanders faring better than Mrs. Clinton against Mr. Trump made it “clear that millions of Americans have growing doubts about the Clinton campaign.”

Throughout the interview, Mrs. Clinton appeared ready to put the primaries behind her and move on to Mr. Trump. Asked if she would consider naming Mr. Sanders her vice-presidential nominee, in an effort to unify the party and bring in his liberal and young supporters, she demurred.

“I think the thing that brings us together is Donald Trump,” she said.

She declined to respond to Mr. Trump’s attacks on her husband, Bill Clinton, including an interview with Fox News on Wednesday night in which Mr. Trump brought up a decades-old rape allegation against the former president. “I think people can judge his campaign for what it is,” Mrs. Clinton said. “I’m going to run my campaign.”

Asked if she had the urge to defend her family’s honor against the onslaught of attacks that will grow louder, Mrs. Clinton said: “No, not at all. I know that is exactly what he is fishing for, and I’m not going to be responding.”

The interview took place in Park Ridge, Ill., the Chicago suburb where Mrs. Clinton spent her childhood. Mr. Cuomo asked what advice her mother, Dorothy Rodham, would have given her in dealing with the likely battle that awaits her against an opponent unlike any she has confronted.

“I think it would be the same advice that my mother always gave me,” Mrs. Clinton said. “Which is everybody gets knocked down and knocked around in life. The real test is whether you get up and keep going.”