Anderson, who suffered the injury in the third Test at Edgbaston, was still around a week away from full fitness, added Cook. He only returned to bowl in the nets on Tuesday.
"Unfortunately he's not going to make it," Cook said. "He's gutted about it. He's made really good progress from where he was at Edgbaston but this is a week or so too soon.
"He bowled in the nets yesterday but at about 60-70 percent and that's not how you go into a game. He's really gutted. The medical team have been brilliant with him but he's still about a week away."
Cook, who added England would field a team unchanged from the one that played in the fourth Test barring a late injury, said leaving Anderson out had been made easier by the fact the Ashes had already been regained.
"Jimmy's not fit," he said. "As much hard work as he has put in, he's not right. It's a hard question to answer, if it was 2-2, but the way Mark (Wood) has come in and bowled in the series, certainly in the first game and the last game, we have options. Especially with Liam Plunkett as well, we have good strength in depth.