Significance doesn't matter, i would contend. Whatever our significance, or lack thereof, in the grand scheme of things, we as individuals know what we value, and we know the concept of fairness, therefore we can understand that getting what we want while not letting others get what they want is not right.

The bolded is the problem, here. Not everyone "knows" the concept of fairness and what is right except what is indoctrinated by society and those around the person. Where does such a knowledge of fairness and value come from in a worldview without supernaturally indocterined morality? 

Most of them still believe in an absolute moral that comes from god. Only what they believe that moral is has changed.

Babies understand fairness

This indicates that we likely have an innate concept of justice, a priori, which does not immediately disprove the idea of God (could lend credence to the concept of the Holy Spirit, that we have been given access to God's capacity for goodness, etc etc), but it does show that humans have a notion of justice without anyone having to explain anything to them, or threaten them with jail, inferior reincarnation, or eternal hellfire.

Only 2/3rds of the babies shared their toys and showed a concept of fairness, and amongst those there were babies that only offered their least favorite toy. That's not a universal, absolute sense of morality. It just shows that morality is varied amongst others as they choose and also to what degree. Even if there is a correlation, having such doesn't give enough justification to dub those that act on them any morally superior than those who don't from an evolutionary standpoint, specifically in cases where going against established morality would benefit yourself greatly.

Well nobody's going to have an absolut sense of morality inherently, certainly, but neither does it need to be supernaturally explained. What the study shows is that the babies have a rudimentary sense of justice, something that, with our rational minds, we can sort out into something more comprehensive and absolute, but that this germ of goodness comes from us (which again, may indirectly be of divine origin) and doesn't need to be transmitted through any sort of scripture, prophet, or messiah.

Well for example.  Sterlyzing everyone who has a genetic disorder now will cause less suffering over the long term as nobody will be born with stuff like taysachs disease.

Or just the outright murder of everyone with aids.  Then no more aids.  Sure your killing people now, but in the future far less people will die.

It's like the old question.  Would you kill one man for the cure to cancer.  You'd have to kill an innocent man but MILLIONS would be saved.

"For the greater good" allows for all sorts of attrocities against the individual.

@RCT

Well obviously if tie your definition of morality to God's character, then of course there can't be morality without him. Basically you're saying "humans lack the ability to be "God" which is quite obvious.

If your definition of morality isn't what's best for the most people possible, then morality is pretty useless. If morality isn't what's best for most people, then why should people be "moral"? We shouldn't. We should do what's best for the most people possible, and if that's not moral by your definition, then humanity shouldn't be "moral". Either way, doing what's best for the most people possible would have the same end result as acting morally.

If you want to take it to an individual plane, if those people do have kids, they are personally responsible for all the suffering of their children, and the generations after them. In that example though, there might be the third option of eventual gene therpay or other treatments that can appear with scientific progress. But I get what you mean, there is also the classic paradox of the hospital with one healthy man and four with severe organ failures, is it moral to kill the healthy man if his organs will allow the other four guys to live?

Such situations might be a flaw in my definition, but remember that the absolute morality championed by different churches is often used to commit even worse atrocities against individuals, for much more arbitrary reasons. I also think, since there can be an actual debate, with real world facts about what is more moral (IE best for the most people) it is the best definition so far. Winston Churchills famous quote of "Democracy is without a doubt the worst form of government, if you disregard all others" comes to mind.

How can we trust our minds to be able to rationally come up with what the "absolute" morals (justice, fairness, etc) are when it is derived from nature being morally neutral and evolving so much over time? Is it from what the majority believes is fair and moral?  It seems odd to say that no one has an absolute sense of morality but claim that absolute moral laws exists. We as humans can try to decide what our morals are and how we live as humans for the most fairness and well-being of everyone, but it cannot be claimed based on that there's an absolute morality and code of conduct that everyone agrees is right or wrong. A natural explanation of the universe involves so much evolution of creatures and survival of the fittest that it can only lead to "morality" being whatever is most adaptable for survival, which really isn't morals or objectively good.