The tourists lost yesterday's fifth one-day international in Adelaide by five runs - their ninth loss in 10 matches down under.
His absence from the three ODIs in the Caribbean is aimed at giving Twenty20 captain Stuart Broad and limited-overs coach Ashley Giles time to prepare for the World Twenty20 in Bangladesh, starting in March.
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Cook will spend some time recuperating from defeats in the Ashes and one-day matches - and plotting how improvements can be made with team director Andy Flower and managing director Paul Downton.
Cook said: "The Twenty20 guys have got their World Cup. They're never together very often and it gives them, and Broady and 'Gilo', time to have six weeks to build for Bangladesh. It's been a real tough tour for me. I am not exhausted, but I'm ready to put the pads away for a couple of weeks or so. I will enjoy seeing my pregnant wife. I'll enjoy seeing a few sheep [on the family farm] for a bit.
Cook plans to use the time away to fine tune the problems that have surfaced over the course of the Australia tour - his last half-century was during the Boxing Day Test.
"That hunger comes back quickly," he said. "I've got a couple of months off where I can do some solid work on my game."
Australia captain Michael Clarke suspected that Cook and his England team-mates were ready to go home.
"There will be a few English players keen to get on that plane tomorrow that's for sure - and keen to spend some time away from the game," he said.
"I know what it feels like to lose a series as a captain. It's certainly tough and you take it to heart. I'm sure Cookie will be looking forward to getting home and having a break. They're a good team and he's a good player. I've no doubt the next time we come up against England they will be as tough as always … in any form of the game. If you take them lightly they'll run over you."
In Adelaide, England had appeared well placed to chase down Australia's total of 217, on a slow wicket, but last man James Tredwell edged Shane Watson behind with two balls remaining after he and Chris Jordan had been left needing eight from the final over.
Stuart Broad and Ben Stokes took three wickets apiece in a bowling performance that restricted Australia to their lowest score of the series. England were on course at 154 for three after a half-century from Joe Root, but lost their last seven wickets for 59 runs to be bowled out for 212.
Ravi Bopara, batting at No.6, had been given out stumped by television umpire Kumar Dharmasena when Australia wicketkeeper Matthew Wade spilled a stumping chance on to the bails. Replays showed Bopara's foot was in the air when the ball made contact with the stumps, although it appeared he may have recovered his ground when the bail fell off.
Victory moved Clarke's team back to the top of the one-day rankings - on Australia Day.
"That's as satisfying a win as we've had recently as a one-day unit," Clarke said. "Winning shows the confidence we have. We still have a few players out resting. We believe we can win no matter what position we are in. We haven't played our best cricket but we've found ways to win."