The Brazilian pushed Barrott after he had been booked by the referee in injury time.
Barrott chose to show Martinelli a yellow card after he was pushed by the Arsenal attacker as he tried to take a quick free-kick.
However, there will be no extra punishment for Martinelli beyond the booking.
The Athletic says that as Barrott took action at the time of the incident, the FA will not be investigating the incident or handing down a more significant punishment.
At the time, ex-ref Mark Halsey said of the incident: "I'm sorry, Gabriel Martinelli showed total disrespect for the match official Sam Barrott in that situation.
"Although Sam issued a yellow card, for me it's a red card. You just cannot put your hands on a referee, let alone push him out of the way.
"You've got to look around the country, up and down, Saturday and Sundays where we have local park referees going out and doing their games. What message does that send out to our local park referees?