You can actually view messages on the mobile web app (or you could earlier today, anyway). On my phone, clicking on messages tells me to download messenger. If I ignore that and hit all messages, it nags again, but eventually does it.

FWIW, I discovered this because Messenger would not show me a group chat that I was added to by co-workers (who I'm not friends with on FB).

Edit: just realized you were referring to the app, not the site. My mistake.