Michael Fogg took to Facebook after reading that funeral directors were refusing to cremate the child-killer, who died on Monday aged 79.
He applauded their stand, saying Brady's 'sick' crimes meant he was not worthy of a proper cremation.
Instead, he suggested the killer's body should be disposed of in the same way police deal with contraband, by throwing it in a council incinerator.
"That seems to me as though all the funeral directors in the North West are reluctant to touch the sick ****. I say good luck to them and ask people to share this post in the hope that it gets back to those folks and show them the country is behind them," he wrote.
The post has been shared more than 7,500 times and received more than 3,000 likes since being published on the Michael Fogg Family Funeral Directors Facebook page on Friday evening.
Brady spent more than five decades behind bars for the murder of five children which he committed with his partner Myra Hindley.
Four of their victims' bodies were found on Saddleworth Moor, but that of 12-year-old Keith Bennett has never been found, with Brady adding to the family's torment by refusing to reveal where he and Hindley buried the youngster.
It has been reported that Brady wished to have his remains burnt and his ashes scattered in Glasgow, where he grew up.
But Glasgow City Council has reportedly said it would refuse any such request and has advised private crematoria to do likewise.