Arnautovic came under fire from Neville and fellow Sky Sports pundit Jamie Carragher for his performance in the Hammers' lacklustre 3-0 defeat to Brighton and Hove Albion last month.
"I respect what Gary Neville achieved as a player. He was brilliant at Manchester United and hats off to him," Arnautovic told Austrian broadcaster Laola1 .
"But what did he do as a coach at Valencia? His job is to criticise, that is 90 per cent of his job.
"I am not saying Gary Neville is a nobody. He has accomplished a lot in his life, he is a legend.
"But as a coach? Let him stay on the TV."
Arnautovic also took aim at his online detractors and promised to prove the doubters wrong.
"Most of them have no work, probably no life and no idea," he said.
"I respect the real fans and I understand them. They were all happy to have me come to West Ham because I had done very well at Stoke.
"But the fans only see what happens in the stadium, they do not see what you do at training and in private.
"I know how strong I am mentally, how I work on myself. I have lost nothing and I'm not in crisis. West Ham can count on me 100 per cent."