It is pretty hard (read impossible) to prove a negative. But it may be that you are really being accused of sharing the blame by being sloppy in keeping your own work private. Some places will treat that as cooperating, even if it is unintentional. If you were sloppy you probably have little recourse.
But one avenue may be open to you (among others). If the written warning is part of your permanent record you can ask to place a written rebuttal into the permanent record as well. If it isn't part of the permanent record, you can probably just move on. If you are questioned about it in the future, assure them that you really weren't at fault. But in most places, a single transgression is unlikely to end your career, especially if the official response was a warning.
Other options that might be open would be regular or ad-hoc judiciary panels that can adjudicate such things, though you aren't guaranteed success here.
And it is also just possible that the other person could take full responsibility for the act, leaving you in the clear.