An unnecessary but common response to this claim is to dispute that homosexuality is unnatural by pointing to the abundant evidence of homosexuality in non-human species. But even if homosexuality were unnatural, this wouldn't make it wrong or unhealthy. If we were to shun everything that is unnatural, we'd be forced to abandon our houses, clothes, medicines, technology, art and much else. Clearly there are some unnatural things we consider good. Likewise, we take earthquakes, tsunamis, snake bites and sunburn to be natural, but not good. And those items on the supermarket shelf proclaiming all natural ingredients like salt, sugar and fat are not good for your health. Whether or not homosexuality is natural or unnatural tells us nothing about whether it is right or wrong, desirable or undesirable. The argument is a non-starter (and for those interested, an example of an argumentative fallacy called the appeal to nature).
Suppose for a moment that for some reason all the other unnatural things that we like are permissible because they have nothing to do with sex, that sex is a special case in which virtuousness always coincides with naturalness (we'd be guilty of a different argumentative fallacy called special pleading, but let's allow it in this case). If this were so, then we might look to the sexual behaviour of other species for examples of how to behave. We could model our behaviour on bonobo chimpanzees, who continually use sex to reinforce social bonds within a group (same sex or not)[source], female praying mantises who eat their mates once they have served their function, ducks who engage in homosexual necrophilia [source], dogs who mount their owners' legs, and so on. Homosexuality and a whole lot of other things would be permitted under this kind of moral philosophy. Not only that, but our guidance would be riddled with contradictions arising from the fact that different species have different sexual practices. For some species, promiscuity is the norm. For others, monogamy is, and so on. We would have to ask what is natural for humans specifically, but if we are contrasting 'natural' with 'man-made', what sense could there be to such a question?
Even if we wish to cling to the view that 'natural' equals 'good' and even if we can ignore the fact that nature seems to make contradictory judgements about what is good and bad, it would still be difficult to argue that homosexuality in humans is any more natural than practices like celibacy, something that few are inclined to condemn. If homosexuality is wrong or unhealthy, this kind of argument simply does not show it.
Note that the claim that homosexuality is immoral because it is unnatural is a part of a more general family of arguments, which also includes justifying promiscuity by arguing that monogamy is unnatural, or eating meat by arguing that vegetarianism is unnatural. As with homosexuality, nature has no consistent attitude towards these things, and even if it did, no moral conclusions could be drawn about these practices (at least on this basis) since 'natural' simply cannot be equated with 'good'.