[itafroma]

Their TOS does not claim ownership over user-supplied content; see section 2, "All materials displayed or performed on the Site[...](other than Content posted by Subscriber (“Subscriber Content”)) are the property of 9GAG, Inc[...]" and section 3, "Subscriber shall own all Subscriber Content that Subscriber contributes to the Site[...]".

The letter does claim that you are using their content, but the use of which infringes on their trademark and copyright. Your use of the user-submitted content would not infringe on their copyright: users were never asked to assign copyright to 9GAG, and in most cases, would not be able to as they were not the original copyright holders to begin with.

If there was a fundamental copyright problem (i.e., the entire basis of your app is copyright infringement due to an ownership claim on all the user-supplied content) and they wanted to shut your app down, they could've just issued a run-of-the-mill DMCA takedown notice to Google. That they didn't seems, to me, that they are fine with your app as long as you remove the infringing elements.

If you were interested in keeping Nine around without fear of 9GAG taking additional action, I'd ask them what content, exactly, infringes on their copyright and have them enumerate what you need to remove. It may be something relatively straightforward, like certain assets or logos that you may be using in your app.