Footage of a frustrated Ronaldo appeared to show him strike the phone out of a young fan’s hand as he left the Goodison Park pitch at full time.
Anthony Gordon’s first half goal gave relegation-threatened Everton three important points while damaging United’s slim hopes of a top-four finish in the process.
United are investigating the incident involving Ronaldo but the 37-year-old publicly apologised to the supporter for what he described as an “outburst”.
In a statement posted across his social media channels, Ronaldo said: “It’s never easy to deal with emotions in difficult moments such as the one we are facing.
“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.”
Ronaldo returned to the starting line-up against Everton having missed last weekend’s 1-1 draw with Leicester City through illness but struggled to make an impact.
The defeat left United rooted in seventh-place, six points adrift of fourth-place Tottenham Hotspur with no games in hand, facing a year with Champions League football.
“If we play like we did here we just don’t deserve it,” said interim manager Ralf Rangnick.
“The players themselves should be eager to play international football, if possible Champions League. In order to qualify for Europe you have to be able to score in 95 minutes in a game like this.”