Walker, who has a vote for the trophy because he won it himself in 1982, wouldn't rule out the possibility that either player could earn his vote eventually but said the off-field issues of each are too much to overcome right now.
"Until [Winston's] act is cleaned up, I can't give him my vote," Walker said. "And I'll say it right now. ... The situation is the same with Todd Gurley. Right at this moment today, I can't give him my vote. Now if he comes back and I see a change, then that vote can be there. But at this given time today, I can't give him my vote."
Walker also criticized Florida State head coach Jimbo Fisher for enabling Winston.
Gurley was suspended indefinitely by Georgia last Thursday while the school investigates the possibility that the star running back was paid to autograph memorabilia.
SI.com's Andy Staples reported last week that someone told Georgia's compliance office that he paid Gurley $400 to sign 80 items on campus one day this spring. After saying throughout the week that Gurley's status hadn't changed, Georgia released a statement Thursday that said there was no "new news" on Gurley.
Winston has faced a litany of off-field issues since last season, when a rape allegation against him emerged that stemmed from an incident in December 2012. After an investigation, police declined to charge Winston. He was cited for stealing crab legs in the spring and was suspended for Florida State's game against Clemson this season for using obscene language.
The defending Heisman winner is currently facing a hearing for a university code of conduct investigation into the alleged rape and is also being looked into for extensive autographs. Florida State said Friday there's currently no evidence Winston was paid for his autograph.
Fisher has maintained Winston will continue to play, including in the Seminoles' game against Notre Dame on Saturday.
- Ben Estes