The result means that the Scot does not have a trophy to show for his ten years in charge.
Gray said that Moyes can only stay at Everton for so long before finally running out of patience with continually threadbare transfer kitties.
"I'd love him to test himself at Everton with a bundle of money at the club that I love and the club that he loves, given similar resources to what Tottenham have had under Harry Redknapp," Gray told talkSPORT. "I'd love to see David given that kind of money, that kind of backing because I have no doubt this club can be successful if it is allowed to grow.
"If it is not then David has to look at who are bigger than Everton in the Premier League. Where could he go to better himself? There aren't many clubs above Everton that want or need a manager at the moment so he is limited to where David could go to better himself.
"If Chelsea and Tottenham are looking for a British coach who has achieved what he can with Everton, and has the experience and the knowledge to take over at a club like that, I absolutely think he could do it, easily."
Moyes took over the Toffees in 2002.