Actually horses do get a version of foot rot. It's called thrush and is very common if they are in muddy conditions. Coppertox or bleach and water will cure it. But without looking at the horse myself I would recommend the vet. To many variables are there to nail it down to one thing.

 

I agree and know what thrush is but its not exactly foot rot like in cattle. But in this situation it seems far more likely that the horse fell on the Ice than developing thrush overnight. Thrush like footrot takes awhile to set in and any one who pays any attention to there livestock would notice it far before it got to the point of not being able to bear any weight on the effected foot.