If objective moral facts exist then they would be facts of the matter regarding an action, behavior, or thought (I will just use 'behavior') to being 'right' or 'wrong'. Morality does not apply to all behaviors, this is an important and key observation.
There is no proof that objective moral facts actually exist. It could be that right and wrong is entirely an opinion held by a being capable of holding such thoughts. But I would like to present my viewpoint on why objective moral facts MIGHT exist, and a brief consideration of what they might look like. PLEASE note that this is NOT a proof of objective morals, it's merely an exploration of how they MIGHT exist on the nature of the concept of Morality itself (as being distinct from other concepts).
First, what I do mean by "distinct from other concepts". If I said Morality was the distance between planets you would instantly recognize that I was NOT actually talking about Morality. So this concept of distance between planets is not part of the distinct concept of Morality. So what IS the distinct concept of Morality? This is the real question we're trying to address. And our problem is that even WE don't really know what we MEAN when we talk about Morality. All we know is that it's something our brains DO, but the details are fuzzy. But we can identify at least some things that are not part of the concept, and I'll show that we can identify some that are. The question then becomes, on the properties necessary for Morality to be a distinct concept - can we then deduce any Evaluative moral statements that must apply (if you removed them then Morality would no longer be a distinct concept).
Any human endeavor that proposes to ask a question about something requires an Evaluative statement upon which to base any measurements (formal or informal). We often only have a very fuzzy idea of what we mean when we ask such questions. We do this about thousands of things and don't really think anything about it. Is it an objective fact that pressing the right buttons on a telephone will let you speak to your friend in New York? Of course it is? What is your yardstick for measuring this? Well, you push the buttons and you hear your friends voice and later on your talk to your friend in person and they confirm, yes that was me on the phone. Something physical happened in the universe that we could measure - but the thing we're measuring is completely arbitrarily defined by us posing the question in the first place. So what you observe & measure is determined by the very Yardstick that you define. What is a second? Is there an objective fact of a second? No there isn't! We measure a second against an atomic process, when it happens 9,192,631,770 times we call that a second, that fluctuation is the objective fact of reality, 1 second is a quantity of that measurement. We used to measure seconds against rotations of the Earth but we found that wasn't actually just measuring time, it was measuring all kinds of other things that made it (very slightly) unreliable! We had a poor yardstick, we found a better one.
So whenever we want to actually measure something, we must must have an evaluative statement that provides the CORRECT yardstick by which to measure it. We can make due with poor yardsticks, but they will give fuzzier answers.
Is it 'hot' out today? Is an evaluative statement but there are a lot of hidden assumptions. 'hot' relative to what? Most often the person asking would be asking relative to the temperature range in which humans are comfortable. And it isn't JUST temperature, humidity would play an implicit role as well, wind speed, and many other factors would all come into play.
Now imagine that I ask "is it wrong to initiate force against another human being"? This is a question that demands an evaluative statement about Morality by which to measure it. Let's look more closely at Morality in detail, and see if we can deduce any facts independent of our mere intuitions about it.
Built-into the very concept of Morality is the necessity that the agent either be capable of performing the action (a deontological/duty moral) or of not performing the action (or avoiding it). Morality simply doesn't apply to falling rocks, the rocks are never considered immoral. If we removed culpability from the concept of Morality it would not be meaningful.
So we CAN identify at least some of the necessary concepts that are part of Morality. Two other important concepts are intentionality and awareness of the consequences.
Someone who has suffered major brain damage may no longer be able to control their actions, if they harmed someone in such a state we wouldn't hold them morally culpable. They would lack the necessary properties that are inherent in the notion of Morality. They would be acting irrationally.
Neurological studies comparing parts of the brain believed to be responsible for empathy show marked differences between dysfunctioning psychopathic brains and normal brains. So I think that empathy is very likely a necessary property to possess to be considered Morally culpable. I'll grant this one is on weaker ground than the others but I think it is not difficult to imagine how this plays a critical role.
What about the ability to learn? We do not consider infants to be morally culpable, they do not demonstrate the necessary levels of cognition and understanding. They don't seem to be able to help what they do. As they grow more capable it seems rather self-evident that they must learn. A being that could not learn, I do not believe, could become morally culpable, any more than the brain damaged.
What about a brain that can learn, but cannot apply that learning to future behaviors? They must be able to apply what they learn to future behaviors.
So there are a number of things that seem to be inherent and necessary to the concept of Morality:
(3) aware of consequences
(5) empathy (and other intact emotions)
(6) ability to learn
(7) ability to apply knowledge to future behaviors
It certainly seems to me that if any of things are missing then the very question of morality would be irrelevant for that object (or being).
So now, let reconsider our question - on these necessary properties of Morality, "is it wrong to initiate force against another human being?"
If we do, a morally sufficient agent would be culpable, they would have intended to cause harm, they would be aware of the consequences (they would know how others are likely to feel and how they themselves would feel if the situation were reversed; as well as what others might do to them if they are caught), they would have had an opportunity to have learned and apply this knowledge to their action. Would a rational being then act in such a harmful way?
The question is, can you remove the Evaluative statement "It IS wrong to initiate force against another human being" from the concept of Morality and leave the concept intact, or is that statement necessary on the facts of Morality itself?
Perhaps you can, perhaps not, but these are the types of statements that would need to necessarily hold given the inherent properties of a distinct concept of Morality.
Now imagine the set of ALL POSSIBLE Evaluative statements that could possibly relate to Morality, if you can remove every single one of them and leave Morality intact then there are no objective morals. If any of them are necessary on the facts of Morality, as a distinct concept, then those would be objective moral facts.
I do NOT think the properties I've listed here are, by themselves, sufficient to sustain any objective moral facts. I've only tried to point how I think that such a thing could exist and what they might look like and how we might eventually discern them. Before we can do any proving, I think that we would need to understand what mechanisms in our brain processes Morality and understand what the necessary properties of those structures are.
I do think that modern neurological studies are showing extremely strongly that there is a (common) neurological basis for human morality, so even if there are no objective moral facts there is almost certainly a phenomenological basis for morality. I'll try to expand on that in the future (or dig up some good resources).