Cahill played for Moyes for over nine years.
"He's done so much for the game, and to have the opportunity to go to one of the biggest clubs in the world and continue their level of success is a remarkable measure of the man he is and how much they respect him as a football manager,'' Cahill told the New York Post.
"No one deserves it more than him.
"Do you want that manager that's been at the top level and bounced around all the European clubs, bringing in all his own staff in, changing the footprint of what Manchester United's all about, breaking wage structures even more, signing the big superstars?'' Cahill asked. "Or do you want somebody that respects the club, respects the production of what they're doing and understands the budget?'