Not according to the article , links, and comments in the article that is the basis for this thread.
"Concentrated tattoo inks may be made from products that were never intended to be used for tattoos. Tattoo ink manufacturers may use products such as calligraphy ink, drawing ink, or even printer ink to make the products eventually used for tattooing. These manufacturers often sell their products online, and while their states may require them to hold a business license, there is no regulation or oversight of the product itself."
However, when problems are identified with any cosmetic prodect, nutritional supplement, vitamin, herbal ect...the FDA investigates after the fact.
"FDA investigates and intervenes when a serious safety issue arises. And that’s what happened here."
Having said that I'm only regurgitating what was in the articles and links provided within the article, I'm no expert on FDA regulation of tattoos. I do know laws, professional licenses, and regulation very by state for tattooists and their buisnesses.