The 25-year-old centre-back was at fault in the buildup to Aston Villa's goal just before half time as striker Mbwana Samatta capitalised on an untimely fall by the England international.
Stones has recently suffered through injury, but since his return he has been left out of several City matches, making just four starts in 19 games across all competitions.
The 34-year-old has since taken to social media to encourage the Englishman to move on.
"Life is full of ups and downs," Fernandinho wrote on Twitter, alongside a picture of the pair hugging.
"Those who are able to get up and move on after they have fallen are much bigger than those who laugh at your fall and point their finger at you. Regardless of what happens, we should always move on... #johnstones".
Stones will be hoping to keep his place in the starting lineup for City's next match in the FA Cup fifth round away at Sheffield Wednesday this evening.