You also might want to check this out: https://www.ted.com/talks/sam_harris_science_can_show_what_s...
The challenge, though, really is not if something is 'objectively right'. That's not the question you want to ask, or the conclusion you really want. You want to know 'is x religion the one true religion?'. We can answer that.
A religion, like anything making a claim, is actually falsifiable. The added challenge with a religion is that the religion has to be 100% right, all the time, or it is not right at all. Said another way, if a religion is ever wrong, it is not true and we can move on from it. This is where religion /always/ gets into trouble. Christianity is no different than any other religion here. It makes claims or assertions and if any of those are shown to be wrong, oops, the religion is not true.
So, when the bible states slavery is fine, rape is fine, beating your wife is fine, it has to be 100% right on these or the religion is null and void. Unlike the global flood (which is obviously not true), you cannot slip up and say "just kidding, metaphor or allegory" in these cases. Making a claim about what is good/bad is something we can judge, particularly as society advances. It doesn't actually matter if those are "objectively right or wrong" for it to make the religion not true, either. That's a trap. You can try and argue that slavery, per the bible, is ok. You can try. You won't win any favor with anyone these days. We've moved past owning people as a concept. It's unacceptable. Our modern moral code, subjective or objective, is "better" than that described in the bible, therefore, the bible is wrong, therefore, christianity is wrong.
One could try and argue that slavery is actually objectively ok, therefore inline with the bible, and some do try this approach and have used it to justify slavery centuries ago. But you can't do that today and be considered a rational human being. It's not acceptable. We know this. It's undisputed fact that owning people is wrong. Therefore, again, our modern moral code demonstrates a higher/better/whatever moral code than christianity in the bible, therefore christianity is wrong and we move on. If a religion were true, society should NEVER be more advanced morally/ethically than the religious standard. This simply cannot be with what the religion is claiming to be.
So, as I stated earlier, the question isn't objective or subjective morality. The question is "is x religion true" (unlike "is there a god?" which really isn't answerable at the moment, maybe ever). We can answer the questions if certain religions are true. To this end, we know the answers on all religions that make positive and exclusive claims...all have fallen short.