The Rasmussen family of Houston says Lulu, their 5-year-old Cavalier King Charles Spaniel, was cleared to fly by a veterinarian days before the flight.
"We are still waiting for answers, but what we do know is that our flight was delayed on the runway in Houston for more almost two hours," the family wrote in a Facebook post.
The family also claims that there were air conditioning problems in the main cabin.
They received the tragic news of Lulu's death when their flight arrived in San Francisco.
In a statement to ABC News, United Airlines apologized for the incident.
"We are so sorry to learn of Lulu’s passing and have reached out to our customer to offer our condolences and assistance. We are deeply upset any time an animal suffers an injury while traveling with us and especially grieved in the rare instance that one passes away. We are conducting a thorough review of this incident," the statement said.
Since 2012, one-third of all animal deaths on U.S. carriers occurred on United Airlines, according to Department of Transportation data.
The Rasmussen family has not said if they will take any legal action against the airline.
ABC News' Whitney Lloyd and Meredith McGraw contributed to this report.