Alaska Airlineshas since apologised and said it is investigating the incident.
"This unfortunate incident was caused by a seating error, compounded by a full flight and a crew seeking an on-time departure and nothing more than that," the airline said.
"It's our policy to keep all families together whenever possible; that didn't happen here and we are deeply sorry for the situation. We've reached out to Mr Cooley to offer our sincere apologies for what happened and we are seeking to make it right."
"Alaska Airlines has a zero-tolerance policy for discrimination of any kind," the airline added.
A spokeswoman did not immediately respond to a follow-up question about what caused the seating error, but the airline told CBS News that it had mistakenly booked two people in one seat.
The airline also said it has a history of supporting LGBTQ rights. For example, the Human Rights Campaign gave Alaska Airlines a perfect rating in evaluating company policies that promote diversity and prevent discrimination against LGBTQ employees. The company's website also has a dedicated "LGBT travel planning" page, a one-stop shop for finding vacation spots, special events and Pride celebrations.
"Diversity and inclusion are part of the fabric of Alaska Airlines," the airline said, adding that it works with nonprofit LGBTQ organisations.
In his Facebook post, Cooley said he and his boyfriend flew home to Los Angeles aboard a Delta Air Lines flight. He vowed to never fly Alaska Airlines again and urged his friends to do the same.
At least one advocacy group has spoken out about the incident.
"Good morning, @AlaskaAir. How is this investigation going? Inquiring #LGBTQ and #Ally minds would like to know!" PFLAG National, a Washington-based LGBTQ group, tweeted.
Alaska Airlines is not the first carrier to face similar allegations of discrimination. Last year, a New York family said they were denied family boarding privileges on a Southwest Airlines flight because the parents are a same-sex couple.
"We approached the . . . 'family boarding area' as we have done many times," Grant Morse, who was flying from Buffalo to Fort Lauderdale, Florida, with his husband, three children and the children's grandmother, told HuffPost.
"The boarding agent assertively approached and said, 'This is family boarding.' My husband responded, 'We know, we are a family.' She said, 'Not all can go. This is family boarding.' "
Southwest Airlines denied that it discriminated against the couple and that the group wasn't allowed to board together because the grandmother was assigned in a different boarding group.
