Mr. Romney, who on Monday called the remarks inelegant, suggested on Tuesday that it was time for a full debate about dependency, entitlements and what his campaign characterized as a long history of Mr. Obama’s support for “redistributionist” policies.

But despite the effort by Mr. Romney to take the offensive, his campaign spent the day working to keep the episode from becoming a turning point in a campaign that until now has remained neck and neck, and trying to minimize the damage from the disclosure of another set of remarks from the fund-raiser, in which he suggested that a two-state solution for peace between the Israelis and Palestinians — longstanding United States policy — was not feasible.

Some Republicans applauded Mr. Romney’s determination not to back away from his statements about taxes and entitlements, which echo themes promoted in recent years by many conservatives, including Mr. Romney’s running mate, Representative Paul D. Ryan. Speaking to a Nevada TV station on Tuesday, Mr. Ryan echoed Mr. Romney’s line of attack on Mr. Obama and government dependency, though he said Mr. Romney’s comments were “inarticulate.” But the developments gave Democrats new ammunition to make their case that Mr. Romney is out of touch with the needs and values of the middle class and does not understand the economic forces at work in many families.

It also left some Republicans distancing themselves from Mr. Romney’s remarks. And it forced the Romney campaign to adopt a new message just a day after starting an ad campaign built around different themes, as officials closely monitored whether donors were growing more nervous about the management of Mr. Romney’s candidacy and his prospects in November.

In an appearance on “Late Show” with David Letterman, Mr. Obama accused Mr. Romney of “writing off a big chunk of the country” and said it would be wrong for a politician to “suggest that because someone doesn’t agree with me that they’re victims or they’re unpatriotic.”