The second-round replay was called off 90 minutes before kick-off due to a frozen pitch at Meadow Lane, with large parts of the playing surface covered in ice.
County had previously assured supporters that the game would go ahead on Tuesday morning, but a severe drop in temperature left referee Darren Deadman with no choice.
Rotherham boss Steve Evans believes Deadman got his postponement decision "100 per cent right".
"The referee wanted to make a decision when both managers were present and he said there was an area of about 25 yards that was hard and icy and dangerous," Evans said.
"He is 100 per cent right. The criticism is could this have been done earlier given the temperatures were not going to get any better. We have got people paying hard money to come down and watch it.
"We prepared properly and fancied our chances, we played particularly well on Saturday and when you play that well you want to go out and play again. The players were looking forward to it and that has to go on to the back burner.
"Darren is an experienced official and if we weren't here we'd probably question it but when you come and see it you know it's the right decision."
The initial tie at the New York Stadium finished 1-1, with Ben Pringle levelling matters for the Millers after Yoann Arquin put County ahead.
No new date has yet been arranged.