The passenger was asked to move the bag because it was sticking out into the aisle, according to a person familiar with the incident.
The owner of the French bulldog puppy told the flight attendant that the animal was in the bag, but the flight attendant insisted that the bag be put in the overhead bin, according to Gremminger.
"By the end of the flight, the dog was dead," Gremminger wrote on Facebook. "The woman, crying in the airplane aisle on the floor."
Gremminger, who did not immediately return a request to comment, posted that she heard the dog barking and "we didn't know it was a barking cry for help."
United has reached out to passengers who were on the flight for more information about the incident.
"We assume full responsibility for this tragedy and express our deepest condolences to the family and are committed to supporting them," said the airline's statement. "We are thoroughly investigating what occurred to prevent this from ever happening again."
The airline is paying for a necropsy of the small dog and is refunding the tickets.
United has faced public outcry after other animals have died on its planes. The airline said it transported 138,178 animals in 2017, more than any other airline, according to the Department of Transportation.
The airline reported to the Department of Transportation the highest number of animal deaths of any U.S. carrier: 18, a rate of 2.24 per 10,000 transported animals. American and Delta each reported that two animals died on their planes last year. Those figures refer to animals that were transported in the cargo hold, not the cabin. United spokesman Hobart said many of the animal deaths were due to animals' preexisting conditions.
Last August, a King Charles spaniel named Lulu on a Houston to San Francisco flight died in the cargo hold after a long tarmac delay. In April, giant rabbit Simon was found dead after a United flight from London to Chicago.
That incident occurred a few months after passenger David Dao was violently dragged off a flight to make room for commuting crew members. United apologized for that incident after an outcry from consumers.
After the passenger dragging, United launched computer-based customer service training. Starting this year, the airline introduced a new curriculum for front-line employees like flight attendants that teaches safety, efficiency and compassion.