After asking flight attendants for cleaning supplies, Mehehan said he was only given "two paper towels and one of those little bottles of Bombay Sapphire".
He told the news station he spoke to an airline manager but wasn't given any further assistance.
"She said to me, 'Well, that's not my problem.' I said, 'I'm sorry?' She says, 'Well, if the cleaning crew didn't clean your seat, I don't have any control over that,'" Meehan told WSB-TV 2.
Passengers were forced to cover the mess with blankets, the station reported.
Meehan described the incident as "dehumanising".
I felt like an animal," he said. "Words can't even describe how awful it was, and they had so many opportunities to make it right."
In a statement, Delta Air Lines apologised for the incident.
"The safety and health of our customers and employees is our top priority, and we are conducting a full investigation while following up with the right teams to prevent this from happening again," a spokesperson told Business Insider.
Upon landing, the aircraft was taken out of service to be deep-cleaned and disinfected.