Vice President Biden shakes hands with an employee of DeNA during his visit to DeNA Co., Ltd, a global Internet company providing web services for mobile devices and PCs, in Tokyo on Dec. 3. (Koji Sasahara/AP)

Vice President Biden’s trip to Asia was the perfect opportunity for a politician with a reputation for verbal gaffes to show off his serious political chops. A skirmish between China and Japan thrust him into the role of international mediator on the eve of his departure.

But no sooner had Biden arrived at his first stop in Tokyo on Wednesday than his aides were greeted with a nightmare series of headlines and cable TV shout-fests back in Washington over whether the vice president is a sexist with old-fashioned, patriarchal attitudes toward women.

The evidence in question was a comment Biden made during a carefully staged photo op with female employees at a Japanese Internet firm. Sliding into a booth next to five young women, Biden opened his small talk with them by asking, “Do your husbands like you working full time?”

Back in Washington, journalists and political operatives quickly made hay over the comment, which was first noted in a pool report by journalists traveling with Biden and then backed up by a video clip from a Japanese TV network obtained by several Web sites.

CNN’s “Crossfire” did a segment in which Newt Gingrich called it Biden’s “war on women”; mega-site BuzzFeed picked up the story; and Republicans linked to the stories in tweets and e-mails. CNN’s Jake Tapper sought to explore on his show, “The Lead,” why Biden’s latest alleged gaffe had become such a big deal.

The firestorm not only distracted Biden’s communications staff from devoting their full attention to the high-stakes bilateral meetings between the vice president and leaders of Japan, China and South Korea, but it also managed to undermine the entire purpose of Biden’s visit to the Internet company — to promote the need for Japan to accept more women into the workplace.

Biden agreed to highlight the issue because the Obama administration, which has made women’s rights a key pillar of its foreign policy agenda, wanted to support Japanese Prime Minister Shinzo Abe’s push for women to augment the shrinking workforce in a nation with a rapidly aging population and a plummeting birthrate. In Japan, an estimated 60 percent of women quit their jobs after having children, in part because companies do not offer flexible leave or work schedules and day-care options are expensive and scarce.

“The most important orthodoxy to challenge around the world is the orthodoxy that says, somehow, women have a limited role,” Biden said while hosting a roundtable discussion on the topic at the Tokyo Internet company DeNA, sitting next to Catherine Russell, the U.S ambassador-at-large for global women’s issues, and Caroline Kennedy, the U.S. ambassador to Japan. “I’ve said this many times in America, and I’ll continue to say it: I personally will not rest until my four granddaughters have every single, solitary opportunity my grandson has.”

That wasn’t just hyperbole. Biden’s granddaughter Finnegan, 15, was traveling with him on the Asia trip, along with her father, Hunter Biden. Finnegan rode aboard Air Force Two and accompanied her 71-year-old grandfather on a public trip to a teahouse in China and a tour of Korea’s demilitarized zone.

“Is she here?” Chinese Premier Li Keqiang asked Biden of Finnegan, breaking the ice as the two leaders met Thursday at the Zhongnanhai leadership compound, the exclusive preserve of China’s senior leadership.

“No, she’s out seeing the city,” Biden replied.

The vice president, meeting with the heads of state of the three nations, had several high-ranking women on his negotiating team: Kennedy, Russell, deputy national security adviser Caroline Atkinson and his deputy chief of staff, Shailagh Murray. But among the three dozen officials from the Asian countries who sat across from the Americans, just one was a woman: Korean President Park ­Geun-hye, whose advisers were all male.

At the Japanese Internet company, Russell told an anecdote about her own experience when she quit her job as then-senator Biden’s staff director to raise her children full time. After 10 years, she called him to ask if he’d hire her back on a part-time schedule.

“He said to me, ‘You come back whenever you want, on whatever schedule you want, because you are worth it to me,’ ” Russell said.

Aides said that behind the scenes, Abe thanked Biden for highlighting the issue, and Biden introduced Russell to Abe and Park. Russell paid separate visits to government officials in Japan and South Korea, talked with students at two women’s universities to learn more about their concerns and ambitions, and met with survivors of violence at two women’s shelters.

In his final meeting, Biden sat down with South Korean Prime Minister Chung Hong-won, who opened the discussion by noting that he had once been a teacher like Biden’s wife, Jill, who has a doctorate in English and is a professor at Northern Virginia Community College.

Biden replied that his wife is the first person in history to continue teaching while serving as the nation’s second lady.

“She often reminds me I have one full-time job and she has two,” Biden joked.

The next morning, aides distributed a transcript of a correction issued by CNN’s Tapper, who apologized to Biden on the air for failing to “provide the proper context” while playing the sound bite of the vice president’s remarks in Tokyo.