Jana Lutteropp, 20, who had been on life support, died at Maui Memorial Medical Center, hospital spokeswoman Carol Clark said.
"Jana fought hard to stay alive," said a statement from her mother and sister, which was released by Clark. "However, we are sad to say that she lost her fight today."
Clark said the family was requesting privacy.
Lutteropp was snorkeling up to 100 yards off Palauea Beach at the resort community of Makena when the shark bit off her right arm.
A high school teacher visiting from California jumped into the water after hearing her screaming and seeing blood in the surf. Rick Moore, 57, of Laguna Niguel, Calif., said Lutteropp went in and out of consciousness and kept repeating that she was going to die.
It's not known what type of shark bit Lutteropp. State officials investigating the attack said witnesses didn't see the animal.
"Jana was a very beautiful, strong, young woman who was always laughing, and we will forever remember her that way," said the statement from her mother, Jutta Lutteropp and sister, Julia Broeske.
They asked that donations in her memory be made to the Maui Memorial Medical Center Foundation.
"We appreciate all the support from the Maui community, as well as the prayers and thoughts from around the world and in Germany," they said. "We especially want to thank the wonderful caregivers and everyone at Maui Memorial Medical Center."
After the attack, Moore's friend Nicholas Grisaffi stood in neck-high water and took Lutteropp from Moore, carrying her limp body out of the water. Grisaffi said Wednesday they've been praying for her.
"I was really hoping it would be a miracle and she would pull through," said the 61-year-old Moore of Laguna Beach, Calif.
Grisaffi said he's been replaying the ordeal in his mind.
"Rick risked his life," Grisaffi said. "Did I do enough? Should I have grabbed my fins and swam out with him?"
The last time someone in Hawaii died from a shark attack was in 2004, when a tiger shark bit Willis McInnis in the leg while he was surfing 100 yards off Maui. McInnis suffered severe blood loss and died on the shore despite rescue efforts by beachgoers, police and paramedics. The last fatal attack before that was in 1992.
Hawaii officials announced Tuesday they plan to spend the next two years studying tiger shark movements around Maui amid what they call an unprecedented spike in overall shark attacks since the start of 2012.
There have been eight attacks statewide this year and 10 in 2012. Hawaii usually sees only three to four attacks each year.