The boy's father jumped in, and the two men began to fight, police said.
"As the father got the upper hand in the fight, witnesses said, the other man pulled a gun and said, 'What the (expletive) are you going to do now?' " police wrote on Facebook.
The father put his arms up and replied: "Nothing. I'm good," according to the police report.
Others in the theater scattered when they spotted the firearm, police said. A few people brought the armed man, who was from Illinois, out of the theater.
Authorities said this weekend that they expected to seek charges against the armed man.
"The man was barred from Cinemark property, and police will present their investigation on Monday to the McCracken County Attorney's office for charges," police wrote on Facebook.
Robin Newberry of the Paducah Police Department said in an email to The Washington Post that authorities were waiting to hear about charges Monday afternoon.
The West Kentucky Star reported that the confrontation occurred during a showing of Star Trek Beyond, though Newberry said that as she understands it, the film either hadn't started yet or had just begun.
Cinemark did not return a telephone message left Monday.
In 2014, one person was killed at a Florida movie theater when a man allegedly pulled out a gun after an argument about texting.