The legendary star was found dead at his home in California on Monday following an apparent suicide by hanging, according to a preliminary coroner's report.
When reacting to the news on Tuesday during his breakfast show, Brazil said that he had 'no sympathy' for the actor, who won an Oscar in 1997 for his role in Good Will Hunting.
He said on air: "I believe he's got a daughter as well, what's she feeling like this morning? Now she's got to sort the arrangements out. I think it's just shocking, I really do. I don't have a lot of sympathy, I'm sorry."
After receiving criticism for his comments, some of which came from Brazil's talkSPORT colleague Stan Collymore, who himself has suffered from depression, the radio presenter took to Twitter.
He wrote: "Well done @StanCollymore for retweeting comments about me. I stand by what I said today. The sympathy I do have is for RW's family & friends."
A spokeswoman for talkSPORT has apologised for Brazil's comments and insisted that his views do not reflect those of the radio station.