The interview was intended to promote an exhibition at the Smithsonian Institution’s National Museum of African Art, which features some works from Mr. Cosby’s collection, but when asked about the furor, he declined to comment.

After the interview, while still being filmed, Mr. Cosby asked that none of the video of his refusal to comment be shown.

“I would appreciate it if it was scuttled,” he said.

In her interview with the local TV station in Florida, Ms. Serignese said that after meeting Mr. Cosby in the hotel gift shop, he invited her to watch the show, and she was given a front-row seat. She said that while she was alone with Mr. Cosby at an after-party, he offered her three large pills and said she should take them. She said she felt that she could not object. She was then sexually assaulted, she said.

After that, she said, she stayed at a room in his penthouse suite for several weeks, but he made her leave when she told him she might be pregnant, which turned out to be a false alarm. They kept in touch, she said, and, later, when she got into financial difficulty, Mr. Cosby sent her money.

Mr. Cosby’s lawyer Martin D. Singer did not reply to a request for comment.

Mr. Cosby’s other lawyers have characterized the accusations that he drugged, raped or molested women decades ago as “decade-old, discredited allegations.”