Everybody, from as early as they can communicate, attempts to justify their actions. That is, they give reasons for why they act a certain way, which reasons are meant to explain why what they did was not wrong or even that it was right. We also demand justification from others, especially when we initially don't agree with what they have done. I think this shows that morality is innate.
Morality has shown to be an evolutionary function. Organisms gain an advantage adapting to their environments when they cooperate for mutual gain. We as human beings tend to have a natural sense of morality because our ancestors adapted to their environment using it and managed to pass along their genes.
Recent studies seem to show that some animals have an understanding of fairness, and fairness is considered one of the first principles of a just society. I don't think much of what we consider moral would be understood by animals because much of our moral distinctions have evolved over time, especially when we started living together in large groups, we had to define rules by which to get along. Animals seem to use instinctual responses to achieve the same purpose so I would think our sense of morality has evolved out of some baser instincts.
I would say that it must be acknowledged that morality, to an extent, is absolutely innate.
I say this because it has been shown in several studies that even babies seem to have an innate sense of fair play and a basic sense of right and wrong. It's also been shown that they have a sense of empathy, something that is the basis of certain items of morality.
Various Studies showed that animals can be breed for a more social/moral behaviour. Obviously there is a strong connection between how "social" an animal, including humans, is and how moral it will act. Cultural differences may be an influence as well, but this begs the question how individuals within a certain group can completely disagree on moral questions
I believe we all have an inherent sense of right and wrong, to some degree. There are certainly nuances of societal rules that we learn, but I believe that if we look to nature we can learn that even animals have a sense of right and wrong. Murder is not an issue in the animal kingdom; most creatures do not kill except in self defense. Perhaps that is because they have a sense it is wrong, just as we do.
We a social beings like all primates. There had to be some early form of morality in proto-humans that carried over to sapiens. We would not be here if it wasn't innate. I have an autistic son, and my wife and I spent his early years teaching him the basics. What we did not teach him how to treat others, or least we had no conscious effort on this aspect. We focused so much time on personal development that it feels neglectful in retrospect. Yet, one day a small child needed help navigating passed an obstruction. Our son, unprompted, went to aid the smaller child.
Psychopaths are those with no sense of morality. They see no issue with using violence to achieve their aims and it has been shown that they have no negative emotional responses to acts of violence or the pain of others which other people do. Psychopathy is caused by damage or a defect in the part of the brain which controls emotions, particularly emotions such as fear and anxiety. This implies that a sense of morality is linked to our emotional responses to certain actions, both our own and ones we see. Therefore, while our sense of morality may be affected by our society, it is at least linked with our emotional capability. This suggests that morality has to be, at least basically, innate.
Morality is innate. Even a child knows what is right from what is wrong. There appears to be some kind of unconscious process driving moral judgments without its being accessible to conscious reflection. But then some people override their sense of morality and carry out immoral acts. In their heart of hearts they must be knowing the wrongfulness of their conduct and after realizing this they would repent. The unconscious process becomes conscious when we contemplate the results of our action or lack of it.
It's there and it wont go away, in forests, fields, rivers, mountains, sky, all pointing to a greater reality and harmony. So we feel grief at death, pain at loss, injustice when betrayed, deceived when lied to. I call it natural theology that points me to a revealed theology which has more specifics in it (being a Christian) but the basics are there for everyone to see and signposts to follow or ignore.
I don't fully believe humans are born as a blank slate but have a basic form of personality. I don't believe morality in essence is innate. The variety of what is considered moral is extremely diverse. I would go so far as to argue morality is a situational tool and system used by humans to cohabitate in large groups rather than a function of being. Morality is a flexible idea rather than a set of governing universal rules. The main problem I believe facing the argument of morality being innate is the belief that actions are categorized as good or bad. Good and bad are subjective ideas based on the needs of society during that particular period or situation. Slavery for example; in modern times it is clearly considered bad, Evil, Unethical, Etc. . . However during the time slavery appeared it wasn't considered as such as it wasn't taught to populations as wrong. Morality can even be argued as a cultural construct as the values tied to morality change vastly based on populations especially if the populations are left to develop on their own without foreign interference.
The Ring of Gyges /ˈdʒaɪˌdʒiːz/ (Greek: Γύγου Δακτύλιος) is a mythical magical artifact mentioned by the philosopher Plato in Book 2 of his Republic (2:359a–2:360d).  It grants its owner the power to become invisible at will. Through the story of the ring, Republic considers whether an intelligent person would be moral if they did not have to fear being caught and punished for doing injustices.
The great thinker Plato was of the view that morality is neither innate nor is it a contingent gift of nature, rather people can be taught to be moral beings. People learn the sense of right and wrong through their experiences. Children are taught to be moral beings through moral education in school.
Plain and simple, morality is a construct created by humans to justify certain behaviors. Some philosophers, like Kant, have claimed morality exists inherently in the universe. However if you dissect history it is apparent that morality is not innate but a system of behaviors designated by certain groups (typically those in power) to control the masses. One need only cite a few examples -- such as homosexuality, automobiles, and/or suicide -- to realize there is no objective "innate" answer. The "morality" of each of these items is based on cultural perspective at certain times in history. Morality is completely subjective and fluid and, therefore, not innate.
Morality is right versus wrong. I have seen nothing that is even remotely accepted universally among humans as to what right is. Cannibals had no problem with the morality of killing other humans. Of course those with morality told them it was wrong. How innate is that? And if you believe in evolution from animals, that would mean animals have morality. Oh wait. Humans evolved into morality. It just keeps getting better. Sorry. Morality is learned. Pets even learn right from wrong if reprimanded enough. Better check who reprimanded you and for what.
It may be innate for people to have a tendency to feel a certain way about certain things. But we live in a real world with real things and it is irresponsible to just say "this is how I feel therefore that's morality". Morality should be based on rational considerations of what will promote human flourishing and our feelings no matter how innate they may feel set aside to do what is right when the situation calls for it.
If morality were innate, there would be no psychopaths or sociopaths walking around, and there are plenty of those. One learns morality from the way one is raised and treated by society, as well as other factors. It's not in the DNA like blue eyes or brown hair that one inherits, unfortunately.