Can listening to sad songs lead to depression? When we listen to a sad song, we may experience sadness ourselves.  Can this sadness lead to depression?
If you are talking about clinical depression as an illness, the direct answer to your question is: I don't know. 

There was a study of the effect of different genres of music on how people felt (https://www.ncbi.nlm.nih.gov/pubmed/27900748). Note the participants only listened to the music for 15 minutes and the study in general is not that powerful.

But if you are talking about making people feel sad or low then maybe. There have been studies that show that negative ruminations can lower people's mood. My thinking is that if the songs lead to negative ruminations then yes it can depress your mood.

There are also some other research suggesting that "sadness" as a feeling can be felt helpful or pleasurable by some people.

It is important to distinguish between sadness and depression. The latter is a clinical diagnosis made by a professional that is generally constant in a person's life, the former is a feeling or emotion that is fluid and can change from one moment to another. 

Also an important question to think about is causation: is it really the songs leading to sadness or sadness leading to listening to these songs? 

Reference:
https://www.ncbi.nlm.nih.gov/pubmed/26257625

https://www.ncbi.nlm.nih.gov/pubmed/29094761