Meehan says Delta flight attendants gave him two paper towels and a bottle of Bombay Sapphire Gin to clean his shoes and suit.

"I noticed an odor when I got on board, but I did not for one minute think there was dog feces on the plane," he said. That changed within seconds of sitting down. "I reached down to plug in my charger and I saw it all over the carpet and on the front of my seat. I immediately jumped up." The dog poop was on Meehan's shoes and the back of his suit pants.

That's how Matthew Meehan describes what happened after he sat and stepped in dog poop while boarding a Delta flight from Atlanta to Miami last week. The incident and Meehan's reaction to how Delta has handled the situation reignites the debate over service animals being allowed on commercial airline flights.

"It was so nasty, I had no way to adequately clean it up," he said. (Meehan has posted an image of his soiled shoe on his Facebook page.)

Delta said the feces was left by an ill service dog on a previous flight.

In a statement to CNBC, the airline apologized for the incident and said, "Delta has issued a refund and additional compensation to the customer impacted by this 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. Upon landing in Miami, the aircraft was taken out of service to be deep cleaned and further disinfected."

That apology is not enough for Meehan. He wants Delta to explain why the flight crew failed to adequately respond to the situation when he first sat and stepped in the feces. For example, he says it took several passengers demanding the mess be cleaned up before a service crew came on board. Even then Meehan says the clean-up was not adequate.

"There was still feces in the carpet and feces under my seat. We took blankets to cover the floor and cover the seat," he said. Delta disputes that it failed to clean up the mess. It says the cleaning crew used disinfectant to clean the area on the floor and seats before the plane took off.

Meehan says Delta has offered to compensate him with up to 50,000 frequent flyer miles and to pay for new shoes and a new suit. Meehan has not yet accepted Delta's offer. He says the airline owes him an explanation for how it mishandled the situation.

"I bought a plane ticket and you made me sit and fester in dog feces for two hours. Shame on you," he said. "I'm a million-miler and this is how you treat me?"