Bradley said the flight attendant argued the car seat was not Federal Aviation Administration approved and issued an ultimatum.
"I was nearly in tears and I said, 'My baby is not going to be safe,' and he said, 'That's it, you're off this plane,'" Bradley said.
She reluctantly complied.
The FAA does recommend children under 20 pounds be in a rear-facing seat.
United Airlines, SkyWest's partner, has apologized for the incident and said SkyWest has spoken with its flight attendant.