One plausible area to look for a role unique to men is that of fatherhood. There is, obviously enough, an intuitive appeal to the idea that only men can be fathers. Of course, it is quite possible to raise questions about this.
One of the first things that needs to be sorted out is the distinction between being a biological father and a father. While most fathers are biological fathers, not all of them are. For example, the father of an adopted child would still seem to qualify as a father, even if he never impregnated a woman. Defining what it is to be a biological father seems rather easy: that is the male who provide the sperm that fertilized the egg.
Of course, a little science can make this a bit messy. For example, imagine sperm engineered and grown in a petri dish. This sperm could be used to fertilize an egg, but it would seem odd to classify the sperm as the father. Perhaps the creator of the sperm would be the father, even if the scientist was a woman or a team. However, let this matter be laid aside, perhaps to be discussed more in comments.
Turning back to looking at the role of father (apart from the biological role), it could be seen as a man’s role because a father is supposed to provide a manly role model and teach the manly virtues to his sons (and presumably teach his daughters that many men lack these virtues).
Of course, this account runs into a bit of a problem. If a father is one who provides the manly role model and teaches the manly virtues, there is a clear need to define what it is to be a manly role model and which virtues are manly. In short, looking at the role of being a father does not seem to help define what it is to be a man. Rather, this seems to be a bit of a backwards approach. Instead, what is needed is an account of what it is to be a man and the nature of the manly virtues. Once those are established, then it would be possible to provide an account of what it is to be a father.
There is the possibility that there are no special manly virtues or manly roles that are unique to males. Thus, non-males could occupy those roles and have those virtues. If so, it would be possible that a woman could be a father (in this sense) or even a machine (such as an intelligent robot). Not to be sexist here, it could also be possible that a male could be a mother (non-biological).
Then again, perhaps there are such roles and virtues. So, as an exercise to the reader, what might these roles and virtues be? Also, which ones would be essential (or at least important) to being a father?