How can I document depression for later reference in grad school applications?
Your transcript should be all you need. Presumably, your transcript shows that you studied at a foreign university, and that things did not go well. If you are back to 100% next semester, it will be pretty obvious that the problems were related to studying abroad. You may then want to write something in your statement of purpose when you apply to assure them that attending grad school won't cause problems similar to what happened when you were studying abroad. This should be an easy sell, especially if the exchange program was in a very far away country and the grad program is in your home country/region.

At any rate, graduate school admissions people certainly do not want to see documentation about your health or personal relationships. They just need a believable reason for why that one semester doesn't indicate you are unfit for grad school, and the transcript showing you studied abroad far away and had some problems should address that. The only place you may run into problems is if your application is automatically filtered out before it reaches the committee -- but no way to avoid that. 

The question posed in the title is a bit more challenging. You are able to attribute your difficulties to having studied abroad. For students who suffer mental health challenges at their home universities, graduate school admission is more challenging. Even in this case, however, I do not think they are interested in seeing medical or other documentation.