Point taken regarding calling them out before the situation is resolved: it would be far more respectful to allow the provider a reasonable amount of time to correct the error before explicitly attacking them. In this situation, I can understand it a little more, as recovering the backups from various internet caches might seem like a time-critical operation, and it would be difficult to ask for help with that without some explanation of what's happened. That explanation certainly could be fairly vague, though. Once the situation is resolved, though, if the resolution is unsatisfactory I think it's fair to say so.
I really took issue with your initial comment because "pointing the finger" has some connotation of assigning blame unfairly or unreasonably; I 100% agree that companies with a culture of blame are poisonous, and that people should err on the side of accepting too much responsibility rather than too little, and do their best to not pass blame on to other people.
There's certainly a line, however, across which I think it's reasonable to call someone else out. Where that line is depends on the situation and your relationship: the bar for doing it within your team is astronomically high (you should basically always deal with those things internally), within even the same company is still incredibly high (likewise), but it's lower when it comes to vendor relationships. Where you draw the line is probably different from where I draw it.
So while I agree that he should have waited, I don't think that publicly expressing his anger with his hosting provider after the fact would count as "pointing the finger" or "passing the buck" or otherwise indicative of a lack of personal responsibility; to me it would be understandable frustration and anger out of having been so dramatically let down by a third-party you were contracting with. And honestly, that sort of negative public publicity is one of the strongest checks we have on companies, be they hosting companies or retail stores or any other type of establishment.
More to the point: even if he did have backups, if they really lost his data and were unable to recover it themselves, I think he'd still be justified in outing their failure publicly after the fact. But again, you're definitely right that he should have waited and given the host a chance to resolve the issue before saying anything.