Your best bet may just be to consult with an employment attorney to explore what your options are, and what type of proof would be required. I suspect, as was mentioned above, it would at least involve written/sworn statements from folks documenting what the TD said to them.
As a TD, the person may feel it's their duty/obligation to provide information on former trainees to potential employers, especially if there are concerns. That being said, if the information being spread is false, it's of course severely inappropriate.
If you have a close/good relationship with your supervisor, you could get their take on whether approaching HR would be fruitful. Or you could see if the folks the TD spoke with would be willing to say to the TD that what they're doing may be inappropriate, as they have doubts about the accuracy of the provided information, and they should probably stop. I don't know how well a person who's spreading false information might respond to that type of direct confrontation, but it's an option.