Yes, you can use a ROC curve for this, though you'll need to do some work. You can't really do it with the discretized version of days -- or rather, you could, but it would be boring. But the raw days measurement is close enough to continuous to make sense for ROC analysis. At a guess, (survival > 1 year) is negatively correlated with (days on dialysis), correct? So for some given number of days , you'd treat as a "positive days result" and as a "negative days result," and similarly as a "positive survival result" and as a "negative survival result." Here is the "gold standard", so, for example, is a false negative. Plot the results for where is the greatest number of days anyone spends on dialysis, and you have your ROC curve.
(Note: I may have the signs wrong, in which case the and signs up above should be reversed. But that's the general idea.)
One thing that occurred to me after I wrote this is that there's no reason to assume that or , so the curve you get out of this might not necessarily run from (0,0) to (1,1) like a traditional ROC curve, for a test where ), does. However, the curve will still be within the unit rectangle, and the area under the curve should still be a valid measure of the power of to predict .