A month before Donald J. Trump was elected president, he and his aides watched his daughter’s coolly composed surface crack open.

Inside Trump Tower, the candidate was preparing for a debate when an aide rushed in with news that The Washington Post was about to publish an article saying that Mr. Trump had bragged about grabbing women’s private parts. As Ivanka Trump joined the others waiting to see a video of the episode, her father insisted that the description of his comments did not sound like him.

When the recording finally showed he was wrong, Mr. Trump’s reaction was grudging: He agreed to say he was sorry if anyone was offended. Advisers warned that would not be enough.

Ivanka Trump made an emphatic case for a full-throated apology, according to several people who were present for the crisis discussion that unfolded in Mr. Trump’s 26th-floor office. Raised amid a swirl of tabloid headlines, she had spent her adult life branding herself as her father’s poised, family-focused daughter. She marketed her clothing line with slogans about female empowerment and was finishing a book on the topic. As she spoke, Mr. Trump remained unyielding. His daughter’s eyes welled with tears, her face reddened, and she hurried out in frustration.