The Canadian star mocked Cudi's struggle with depression and suicidal urges on the recently-released Two Birds, One Stone song.
In his lyrical jab at Cudi, who recently checked into rehab, Drake spits, "You were the man on the moon/Now you just go through your phases/Life of the angry and famous".
In another line, he refers to anxiety medication Xanax and painkiller Percocet, adding, "You stay xanned and perc'd up/So when reality set in you don't gotta face it."
Drake's attack fell flat with some fans, who criticised the rapper for making light of such serious issues, and now one angered activist is urging members of the public to make their voices heard by signing a new Care2 petition to pressure him into making amends for his insensitive remarks.
"Mental health is not a joking manner," petition creator Paige D wrote in the post. "According to Mental Health America, the most common disorder amongst suicide victims is depression. Depression or bi-polar disorder affects about 30-70 percent of suicide victims."
The petition included a letter to the Hotline Bling hitmaker, which read: "Dear Drake, I strongly urge you to donate the proceeds from your song, Two Birds One Stone, to mental health charities. This is your opportunity to bring awareness to mental health issues and act as a good role model for your fans."
The goal of 5,000 signatures had already been reached by Thursday morning (03Nov16), prompting Paige to increase the figure to 6,000.
Drake has yet to comment on the campaign.
Cudi, who originally sparked the conflict by blasting his former mentor Kanye West and Drake on Twitter in September (16), took a break from rehab to hit back at his rap rival last week (28Oct16).
"Say it to (my) face, p**sy," he fumed, in a now-deleted tweet. "You think it's a game. I wanna see you say it to my face. I'll be out soon. Promise."