There is a tier of definitions of 'fatherhood', with different levels of naturalness. Sorry for the length, but to make my point I think I need to defend five separate definitions of 'father'.
1) Clearly fathering children in a biological sense comes natural to all male mammals (with a few notable exceptions), and no less so for humans.
2) Males among many of the primate species close to us do not actively take care of their own children, but they do keep track of which children are likely to be theirs, and they ease their way. They also defend them and to some degree mourn their loss https://www.youtube.com/watch?v=Fs-6SKfh4fM Some go to the extreme of surreptitiously killing off children that are not related to them, especially if those competing children come from the same mother, and are therefore competing with his own children for her resources. In some species, this goes beyond tricking the mothers and takes the form of a pointed vendetta. https://www.google.com/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#
Male primates of various also do seem to enjoy the young, to readily defend them, and to mourn their loss. But I do not think this is associated too strongly with their own young over others.
It would be odd if humans did not have similar habits from early in our species' life. To some degree keeping an eye out for the success of your own offspring is an indirect form of fathering in a more social sense.
3) But what is natural to humans has usually been sought by looking at our past. For a long time it was presumed that fatherhood arose rather late in our history because of societies that run on the "Welsh model". There are cultures where inheritance between men flows from mother's-brother to sister's-son, and women avoided sex until their mothers produced brothers, or the adopted men as brothers to defend them and their children.
This pattern is clearly presumed in early Welsh oral traditions that got folded together into traditional story-cycles and recorded as the "Mabinogion". Having a model so close at hand of a culture that did not recognize fatherhood at all, many European thinkers in past centuries have assumed that this represented the natural state, by a sort of Occam's razor: Social conventions tend to come into existence and then stay relevant, and not totally disappear only to be re-invented later. Engels incorporated this view into his class model https://www.marxists.org/archive/marx/works/1884/origin-family/ch02c.htm
But it is more likely, given modern genetic data about how many early men vs how many early women successfully reproduced, that the more normal form of early human social organization was one of collective polygyny, where men had either several wives, or none. (Thus almost every woman reproduced, and a a fraction of the men.)
There, a given man, perhaps a surviving grandfather with a long memory, would have ruled over an extensive land area with his many descendants, and perhaps those of his siblings, providing the majority of his workforce. Such systems still exist today, in places where it requires a lot of land to be cared for very intensively for production to support a population. Fatherhood, as a sort of benevolent monarchy is definitely a strong force in those societies today, and it is difficult to imagine how it could not have been so if this were the natural organization of human labor early in the history of agriculture.
If we step back further, to pre-agricultural societies, we can study the Amerind societies that are based around hunting and gathering in a way that matches what we surmise from fossils of our pre-agricultural ancestors. Such societies have a strong need for active apprenticeship of young hunters. When we look at surviving tribes today, we see that such apprenticeships are often tied to genetic lineage, with most boys apprenticed to fathers or uncles, and fatherless boys 'married' to gay mentors. This suggests a sort of fatherhood that occurs naturally, but begins relatively later in life, just before the start of sexual maturity.
All told, the traditional view that fatherhood is a late-arising social convention related to war, seems to be based upon an odd minority, and most other very ancient ways of life do involve some sort of fatherhood, though nothing anywhere near as potent as motherhood.
4) But there is another layer built on top of the natural connection to children which Lacan argues is an aspect of necessary early experiences we all have and not our biology. As refined by object-relational thinkers in psychology, the basic idea is that the relation of fatherhood creates in us the network of understandings around the concept of belonging and ownership. We belong with/to our mothers in a very natural way, but at some point we realize there is a larger unit to which we belong, and we have obligations to that unit that may go beyond compliance. This comes when we fully understand that our father is part of our family. Then this causes us to re-create and to similarly shape the same experiences in our children.
This can be seen as equally 'natural' even though it is an aspect of the full cultural environment. Each infant would reach this understanding in any society, or the society would not hold together. But our nuclear families with a slight distance between mothering and fathering shape it in specific ways in our shared consciousness.
5) On top of this psychological notion of fatherhood, we have built a layer of fatherhood which is, in some sense, fully artificial. We have projected this notion of family upward onto the tribe and nation, to which we belong in a much more complete way, lessening the tribal identity of women so that they can cross boundaries and leave their tribes to join their husbands'.
And the requirements of the nation have in turn been reflected back on fatherhood itself. This is the patriarchal construct that Marxist-feminists claim is the initial root of classism, and on which men's-rights activists blame things like forced military conscription.
Fatherhood in modern societies is shaped for cultural stability. If a man lends support to a woman and this becomes established to a recognized level, the state considers her children to be his. Fatherhood is bestowed by acceptance of an obligation, and not by biology.
We have the competing notion of establishing paternity via proof, but in some sense this creates a formal support obligation with none of the other aspects of fatherhood. (For instance, the biological relationship of a rapist to the resulting child may be one of obligation, but actual fatherhood is generally not workable. Rapists use this to manipulate their way back out of the paternity obligation by demanding visitation that they know the court will not allow.)
This form of fatherhood, manipulated for State benefit, one can legitimately call artificial. By tying up all of the others to it, we turn them into something that is primarily an obligation, which creates a lot of artificial resistance to becoming a father or to actively fathering. This ties up with our notions of work, particularly men's work, and our strange obligation to dislike it, and to shape it in rigid forms that are easy to dislike, so that we feel OK getting 'compensated' for doing our natural duties.
Failing to separate reproductive, biological, historically natural, psychological, and patriarchal versions of fatherhood in our thinking creates a range of problems of understanding, and leads a range of people to want to relativize or disown the concept or use it to enforce conventions that are not logically connected to it.