Footage emerged of the Manchester United superstar appearing to knock a mobile phone out of an Everton supporter's hand as he walked down the tunnel at full-time following his side's 1-0 defeat on Saturday.
Man United confirmed the club were aware of the matter and looking into it.
A club spokesperson said in a statement: "We are aware of an alleged incident after today's game at Everton and the club will cooperate with any police inquiries."
And on Saturday evening, Ronaldo posted a statement on his personal Instagram account to issue an apology.
He also invited the fan involved in the incident to watch a game at Old Trafford "as a sign of fair-play and sportsmanship".
"It's never easy to deal with emotions in difficult moments such as the one we are facing," he wrote on Instagram.
"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."
Ronaldo had returned to United's starting line-up and played the full 90 minutes of the defeat.
Anthony Gordon's deflected strike decided the match at Goodison, likely condemning United to a season without Champions League football.
The club are now six points from Tottenham who enhanced their chances of finishing fourth in the Premier League by beating Aston Villa.
United goalkeeper David de Gea called his side's display a 'disgrace' post-match, and manager Ralf Rangnick agreed, but it was Ronaldo who made the headlines for the off-field incident.
The legendary goalscorer had already limped off the pitch following the final whistle, sporting a nasty gash on his shin which was picked up by the cameras.
He looked in pain as he entered the Goodison tunnel to head to the dressing room when the altercation with a supporter occurred.