Going back a bit, but the Everton fans thought the same about Howard Kendall, and the Aston Villa fans thought exactly the same about Ron Saunders yet within a year of the fans turning on them they both won the League, and a time Liverpool were dominating the League(and if you don't believe me about Liverpool check out the History channel, be warned a lot of the content is in black'n'white)as well.
Not suggesting for a minute that David Moyes would have won the League the following season if he'd stayed on, but there is no reason why once he'd rooted out the rotten apples(Hi Rio) that he couldn't have turned things around, and quite quickly.