As "chief prosecutor" here, I'm in the delete camp, and wanted to explain why.
Ordinarily, disclosure is great. If someone comes and provides a useful answer, and says "btw, this is my app; let me know if you have any feedback or comments," that's a helpful answer and should be encouraged. The SE rules explicitly allow for this kind of behavior, unlike some online communities that ban all self-links (and generally ban the users who make them too). This is a good thing. But that's not what happened here. This user breezed in, dropped links to his own app on two questions, and split, without ever explaining he had any connection to the app. He's demonstrated no interest in participating other than self-promotion.
Most people wouldn't tolerate this behavior offline. What would you think if you asked your neighbor to recommend a plumber, he said "oh I know this great guy, does excellent work at a good price," you call him, and a few hours later your neighbor rings your doorbell wearing overalls and a hat, carrying a pipe wrench? I suspect you'd be peeved that he conveniently failed to mention the fact that he was recommending himself. Even if he turned out to be a good plumber, you likely wouldn't get past the fact that he omitted this material information from his recommendation.
Spam destroys communities. What some are proposing is that we just add a note to the answer and move on. But how would that work for other similar situations? We get tons of answers of the form "need a visa? go to myvisahqtotallylegitvisaservice.xyz" that are deleted as spam without a moment's thought. Should we really just all a disclaimer to these answers and keep them?
Can we catch everyone who posts self-links without disclosing that fact? Of course not. Had this user not used his real name and picture, it may well have slipped by (though obviously the tone of it was suspicious enough for several of us to investigate, which is something I don't exactly make a habit of here). But when someone is clearly only interested in using the site for self-promotion and blatantly fails to mention that fact, delete the answers.