You have helpfully sought to explain your terms:
Fairness is impartial and just treatment or behavior without favoritism or discrimination. Ethics is concerned with what is morally good and bad and morally right and wrong.
It would be useful to know how you distinguish the fair from the just. The terms 'fair' and 'just' are sometimes used interchangeably but this is not the case for you since, by your account of fairness, justice is not identical with but an element in or a component of fairness.
A natural context for fairness is the distribution of a good. Suppose we say, then, that fairness in the distribution of a good involves the impartiality of treating like cases equally, different cases differently, and different cases differently in proportion to the extent of their difference. Justice could then be introduced to supply the criterion to be used in determining when, for any given purpose, cases are alike or different.
For instance, we might decide that cases are alike or different on the basis of need or desert. Fairness would then be a matter of treating like cases of need equally or like cases of desert equally, different cases differently, and different cases differently in proportion to the extent of their difference.
So much for fairness. Whether any such distribution is morally good or bad, morally right or wrong, depends on your ethical theory. Suppose, for example, that one adopts a teleological ethical theory according to which the good ought to be distributed in whatever way maximises overall benefit.
There is no a priori guarantee that a fair distribution will yield a distribution that maximises overall benefit. An unfair distribution might maximise overall pleasure or happiness.
I am not recommeding any such ethical theory - or indeed any ethical theory - but simply want to describe a situation in which, depending on how one understands fairness and which ethical theory one adopts, the fair can fail to be the morally good or right - that it can be, in your language, 'actually unethical'.
J. Broome, 'Fairness', Proceedings of the Aristotelian Society, 1990 - 1991, New Series, Vol. 91 (1990 - 1991), pp. 87-101.
H. L. A. Hart, The Concept of Law (London 1961), p.156.
M.N.L. Nathan, 'A Difficulty about Justice', Mind , Apr., 1971, New Series, Vol. 80, No. 318 (Apr., 1971), pp. 227-237.