Head coach Stuart Lancaster confirmed on Monday that the forward would take no further part in the series.
Hooker Dylan Hartley will take over as captain for Saturday's clash in Port Elizabeth.
Lancaster revealed that Robshaw had suffered a blow to the hand in the first Test in Durban before taking another knock in Saturday's second Test loss in Johannesburg.
Lancaster said: "Chris, as you would imagine, is devastated.
"He has led us superbly in the last seven internationals and been one of our standout performers.
"But the medical advice is that he should not play and we have to trust that expert opinion.
"Dylan Hartley will captain the side. He has been a key part of our leadership group and I know he will step up to the task on Saturday."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.