Video footage appeared to show Alves rubbing his nose with his fingers and then wiping them on Ronaldo's sleeve.
The pair had a running battle during Madrid's 2-1 win at the Parc des Princes, with Ronaldo fortunate to avoid punishment after aiming a petulant kick out at Alves.
"In my job it's not painful to be eliminated because that's part of day-to-day life, but to have to see and read stupid things like this..." Alves wrote on Twitter.
"I will continue competing like I have done and leave the nonsense to stupid people #respectwinorlose"
Penoso en mi profesion en la mayoría de las veces no es quedar eliminado, eso forma parte del cotidiano nuestro, pero tener que ver y escuchar bobadas como esa.... pero bueno, continuaré competiendo como hice hasta hoy y las bobadas para los bobos!! #respectwinorlose !! https://t.co/XdHQyesSXm— Daniel Alves (@DaniAlvesD2) March 8, 2018
Alves also apologised for comments he made following the shock death of Fiorentina captain Davide Astori.
"Every day in the world, millions of children die and they probably don't have such big repercussions," Alves said prior to facing Madrid.
The Brazilian was criticised for seemingly showing a lack of compassion, with Nice striker Mario Balotelli among those hitting out at Alves for insensitivity.
"What he said wasn't wrong, but he could've kept it to himself at that moment," Balotelli said on Instagram.
"He should've just passed on his condolences and that's it, not talk rubbish. You cannot make those comments when faced with someone's death."
Se o o Balotelinho falou tá falado, não tive intenção de ofender a ninguém, vivo em mundo que temos que ser hipócrita muitas vezes para que as pessoas gostem de ti but sorry ma people.... enfim, se alguém se sentiu ofendido, desculpaaa. https://t.co/L03x0HHZ39— Daniel Alves (@DaniAlvesD2) March 8, 2018
Responding to Balotelli and his critics, Alves posted on Twitter: "If that's what Balotelinho said I wasn't trying to offend anyone.
"We live in a world where you often have to be hypocritical to be appreciated by people. Anyway, if anyone was offended then I apologise."