A video shared online shows the moment Bishop Charles H Ellis III appeared to touch the singer inappropriately as he hugged her on stage after her performance of Franklin's "(You Make Me Feel Like) A Natural Woman" .
His arm is seen to be around Grande above her waist, with his fingers pressing against one side of her chest.
More than 90,000 tweets with the hashtag #RespectAriana circulated the web with viewers expressing their outrage.
The bishop has since apologised for how he touched the 25-year-old singer, in an interview with the Associated Press, saying it would "never be my intention to touch any woman's breast".
"I don't know, I guess I put my arm around her," he said. "Maybe I crossed the border, maybe I was too friendly or familiar but again, I apologise.
"I hug all the female artists and the male artists," he continued. "Everybody that was up, I shook their hands and hugged them. That's what we are all about in the church. We are all about love."
He added: "The last thing I want to do is to be a distraction to this day. This is all about Aretha Franklin."
He has also admitted his joke about mistaking her name on the programme for a new item on the menu at Mexican fast food restaurant Taco Bell, was inappropriate
"I personally and sincerely apologise to Ariana and to her fans and to the whole Hispanic community," the bishop said.
"When you're doing a programme for nine hours you try to keep it lively, you try to insert some jokes here and there."
Franklin, died aged 76 earlier this month of pancreatic cancer.
Stay up to date with the latest news stories from a Christian perspective. Sign up to our daily newsletter and receive more stories like this straight to your inbox every morning.