SEOUL—Even inside South Korea, few people realized it remains the only country where inking tattoos is illegal without a medical license. But then the obscure law crossed paths this week with something not so obscure: the K-pop band BTS.

Pushing for more relaxed industry regulation, a South Korean lawmaker posted a photo of a BTS member named Jung-kook, who on a local television show had covered up his finger tattoos with bandages.

“Remove Band-Aids from BTS’s body,” wrote the lawmaker, Ryu Ho-jeong, on Twitter.

The backlash from BTS fans was swift, accusing Ms. Ryu of dragging the seven-member band into politics. Ms. Ryu, a self-proclaimed BTS fan herself, has since apologized.

Using a BTS member as an example is part of changing prejudices about tattoos in South Korea and attempts to create a globally competitive tattoo industry in the country, Ms. Ryu said through an aide. Highlighting the issue has generated more attention for giving South Korean tattoo artists legal status, the aide said.

Big Hit Entertainment, the music label representing BTS, declined to comment. The Korean pop group has tens of millions of fans globally, with songs that have topped U.S. charts and hit billions of YouTube views.

Jung-kook, shown in May, had covered up his finger tattoos with bandages on a local television show.

Jung-kook, shown in May, had covered up his finger tattoos with bandages on a local television show.

Photo: Yonhap News/Zuma Press

It isn’t illegal to have tattoos in South Korea. Those inking tattoos without a medical license are technically subject to a fine of up to $950. In socially conservative South Korea, tattoos are often considered taboo, leading some celebrities to cover them up on local TV shows and in media interviews. Until recently, men with “excessive tattoos” could be exempted from mandatory military service.

South Korea exists in a unique place globally, with a full-fledged law limiting tattoo inking still on the books, local lawmakers say. Tattoos are regulated in Islamic countries such as the United Arab Emirates and Iran, which ban profane images. Sri Lanka has deported tourists for having tattoos of Buddha. Japan had a similar rule requiring medical licenses for tattoo artists, but the country’s Supreme Court overturned it last year.

With local police lightly enforcing the tattoo rules, political efforts to abolish limits on South Korea’s tattoo artists haven’t been successful or even garnered much attention, until BTS unknowingly got involved.

Now, Ms. Ryu, who had sponsored a bill seeking to liberalize the tattoo industry, faces new critics like Jane Lee, a 24-year-old university student and BTS fan. Ms. Lee, who got her first tattoo in Seoul last month, hadn’t realized tattooing limits even existed.

Ms. Ryu “used BTS to get political attention for her new bill without any consent from BTS members,” Ms. Lee said, “which is why we were outraged.”

South Korea’s regulations originate from a 1992 Supreme Court ruling that designated tattooing a medical act. At the time, tattoos were associated with criminals and gangsters, and the social consensus was that tattoos were offensive, which created strict regulations, South Korean tattoo artists said.

Efforts over the past decade by lawmakers and tattoo artists to reverse that rule have proven unsuccessful. The main opposition comes from the Korea Medical Association, South Korea’s largest physicians group. Letting those without medical licenses give tattoos could lead to serious infection or allergic reactions, the group says.

Nonetheless, about a quarter of South Koreans in their 20s and 30s have been inked, according to a 2019 report by the Korea Institute for Health and Social Affairs, a government-funded think tank.

Despite the rules, South Korea is home to many celebrity tattoo artists, who have formed a labor union that seeks to abolish the 1992 law.

Kim Do-yoon, the union’s 42-year-old founder, has tattooed Hollywood actors such as Brad Pitt and Steven Yeun. Earlier this year, Mr. Kim became a rare subject of enforcement under the country’s tattoo law. After he uploaded a video of himself inking a tattoo on YouTube, an anonymous person reported him to police. Mr. Kim has since challenged the fine in court, with a trial scheduled to begin next month.

“We get artist visas when we work abroad, and the size of the industry in South Korea can’t be ignored anymore,” Mr. Kim said. “Despite the surge in people getting tattoos, the country still pretends like we don’t exist.”

Write to Dasl Yoon at dasl.yoon@wsj.com