Jan. 5, 2014 -- The family of Olivia Marie Coats laid the infant to rest this weekend, just days after she was removed from life support.

Olivia lived for five days after her parents allege a forceps delivery crushed her little skull and caused brain death.

Now, the family has launched a Facebook campaign to stop the use of forceps in all births.

Allen Coats, 25, and his fiancée Rachel Melancon, 24, say they will sue their obstetrician, Dr. George T. Backardjiev, but not The Medical Center of Southeast Texas, where their daughter was born on Dec. 28. The baby was transferred that day to Houston's Children's Memorial Hermann Hospital, which confirmed the baby died on Jan. 2.

On Saturday the family held a service to celebrate the short life of Olivia and afterwards held a balloon release in honor her.

In a Facebook post attributed to Olivia's parents Allen Coats ,25, and his fiancée Rachel Melancon, 24, called the balloon release "the most beautiful thing ever."

"We are so so so so happy with the turnout of the funeral. So many people came and donated and gave us support," wrote the couple. "We are standing here at peace because all of you! Y'all are truly carrying our pain in y'all's heart."

While Coats and Melancon are still grieving, they have also decided to take action in the hopes that no parents have to go through the heart break that they have. The couple is planning to sue their doctor for using forceps on Olivia in addition to finding a way to pass "Olivia's Law" that would ban forceps from deliveries.

"We're not mad at the hospital, this is not their fault. It is one man's fault," Angie Coats, the baby's grandmother told ABCNews.com. "We only want justice for Olivia; we want the person responsible, which is the doctor. We don't want the hospital being shunned. The hospital is great. The nurses were wonderful. It's not their fault."

Angie Coats said the young parents were devastated by the loss. She said the parents had repeatedly asked the doctor to perform a cesarean in advance of the Christmas Day due date because of the large size of the baby, but he refused.

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"Rachel had a normal, healthy pregnancy and the day she went in to see if she could be induced, it was already after Christmas," she told ABCNews.com. "The baby was so gigantic inside of her. She asked, 'Can you please give me a C-section? This baby isn't coming out of me.'"

Olivia weighed 7 pounds, 14.9 ounces and 22 inches long at birth.

Angie Coats, 43, of Bridge City, Texas, said her daughter-in-law was "itty bitty" -- 4-feet-11 and 95 pounds -- before she was pregnant.

"[Backardjiev] said, 'No, you don't want a C-section. You'll have a scar," said Coats, who was present at the birth. "During her delivery, the baby's heart rate kept going up. He said, 'One more hour, one more hour.' Her water broke, but it was 18 hours until the delivery. [Rachel] was running a 103 fever... Five hours passed, then he came in and she started to push. But she was so worn out and the baby wasn't even in the birth canal."

Coats alleges that the baby was face up and Backardjiev tried unsuccessfully to turn her with his hands.

"When he couldn't do that, he took the small forceps to try to pull the baby out. He kept going and even put his foot up on the bed trying to pull," she said. "He was turning and twisting and she would never come out. He put the forceps one way and the other. When he touched the top and side of the skull, we heard a pop, like clay cracking in pottery and heard her skull crush."

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Olivia was ultimately delivered by emergency C-section, according to the family, but they allege it was too late. "My son said, 'I don't think my baby is alive.' She was not breathing and she never cried," said Angie Coats.

She said the medical staff told the family the baby was alive and would be transferred to Hermann Hospital.

But once the family arrived at Hermann, they allege that doctors told them Olivia was not breathing on her own and had suffered "numerous skull fractures."

ABCNews.com twice called the office of Backardjiev and spoke to an assistant, asking for comment. But the obstetrician did not return calls.

Matt Roberts, the CEO of The Medical Center of Southeast Texas, said in a statement that the baby's death "rips at our hearts and words are insufficient to express how much our sympathies go to this loving family."

The statement added, "While patient privacy and peer review restrictions prevent the hospital from commenting specifically, the hospital administration and independent medical staff immediately initiated a review of all aspects of this case. Our independent medical staff leadership shares in the hospital's commitment to take all necessary actions to understand why this happened."

A spokesman for Children's Memorial Hermann Hospital said Olivia was transferred by Life Flight to its neonatal intensive care unit on the day of the birth. The hospital expressed its "deepest sympathies" to the family, but said patient privacy rules prevented it from commenting on the case.

Angie Coats said that the family was by Olivia's side at Hermann Hospital at her death.

"They took her off life support and it took two to three hours for her heart to stop as my son lay in the room," she said. "I gave her a bath and they took her down to the morgue."

The family has urged others to call for an end to forceps deliveries nationwide. So far, their Facebook page has garnered 55,000 "likes."

ABC's Dr. Jennifer L. Ashton, an obstetrician and gynecologist, said that a forceps delivery can be risky.

"Forceps are used less often today versus vacuum extraction, but skull fracture is a known risk of this procedure," Ashton said. "Often times there is no other option when delivery needs to occur and even a C-section could be not an option."

But Ashton said forceps should not be outlawed. "In the right hands they can save a baby's life," she said. "It takes a lot of skill and practice to perform a forceps delivery and most younger [obstetricians] have been trained in vacuum extraction."

Olivia's parents decided to donate the baby's heart valve to save another baby and her tissue behind her legs and knees was donated for skin grafts. "Something good will come of this," said Coats.

"They are doing great," she said of her son and his wife. "I have never seen so much strength and courage in two people in my whole life. All I hear is they want to help someone else. They want to help other parents and make sure this never happens again."