Footage, taken by @EvertonHub, shows Ronaldo lowering his right hand towards the ground before what appears to be a phone colliding with and bouncing up off the surface, as he exited the pitch following the 1-0 defeat at Goodison Park.
United are aware of the footage, which is being widely shared and discussed on social media following the game, and the club said on Saturday afternoon they were looking into the incident.
Ronaldo said on Instagram: "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 apologize 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"