A video was shared on social media alongside comments claiming he smashed the device as he left the pitch after a 1-0 loss at Everton in April.
Ronaldo, 37, later apologised on social media.
He was cautioned by police regarding the incident in August.
"It is alleged that the forward's conduct after the final whistle was improper and/or violent," the FA said in a statement.
Manchester United said they would support Ronaldo in his response to the charge.
The defeat at Everton dealt a blow to United's hopes of qualifying for the Champions League. They went on to finish sixth.
In an Instagram post after the incident, Ronaldo said it was "never easy to deal with emotions in difficult moments such as the one we are facing".
He added: "Nevertheless, we always have to be respectful, patient and set the example for all the youngsters who love the beautiful game.
"I would like to apologise for my outburst and, if possible, I would like to invite this supporter to watch a game at Old Trafford as a sign of fair play and sportsmanship."
This is an unprecedented case and regarded as non-standard, so there will be a hearing whether Ronaldo accepts the charge or not.
It could result in anything ranging from a warning to a suspension.
Tottenham's Eric Dier was banned for four matches two years ago when he went into the stand to confront a fan who had been in an argument with a family member.
It is an issue Ronaldo could have done without after a difficult summer during which he failed to get his desired move to a Champions League club and started the season on the bench under new manager Erik ten Hag.