His fingers and forearms were paint stained, blacks and reds deep under his nails, flecks on his jeans and boots, something his wife, Melissa Cohen, whom he married in May after a weeklong courtship, can’t stand. “I always get my pants dirty,” he said. “I don’t even notice it, but Melissa hates it.”

Ms. Cohen, who is pregnant with the couple’s first child, is a South African-born filmmaker in her early 30s who lived in Israel for several years. The day after their first date, Mr. Biden tattooed the Hebrew word “shalom” — for peace — on his left bicep, giving the couple matching tattoos. (He also has a tattoo of New York’s Finger Lakes region on his back, because his mother grew up there.) “It’s very abstract, sometimes very dark,” Ms. Cohen said of her husband’s art. “It draws a lot from nature.”

Mr. Biden set out late last year to find gallery representation with the help of Lanette Phillips, who is not a typical artist’s agent. She’s a video producer who ran a management company with clients including Quentin Tarantino and Darren Aronofsky. A longtime Biden family friend, she hosted a star-studded fund-raiser for the former vice president last November at the Pacific Palisades home she shares with Rick Lynch, a partner in an entertainment marketing firm. Mr. Biden did not sign with a gallery, and Ms. Phillips said this week that she is no longer advising him, but Mr. Biden is still setting his sights on exhibiting his work.

Normally, the art world is a fine place to make the right friends, attend the right parties, a venue more forgiving than Washington, D.C. But Mr. Biden could have a challenge convincing the public that the word “artist” belongs in his CV, coming after his careers as a lawyer, lobbyist and director of private equity firms.

He has no formal training as an artist, but he said that he has sketched off and on since age 7, and that he wasn’t dabbling: “It’s something I’ve taken seriously for a long time but hasn’t necessarily been for public consumption.”