Ethical relativism is the view that the truth of ethical statements - such as abortion is wrong - is dependent on, or relative to, the culture, group or individual making the statement. Ethics is relative to groups, cultures or times.
But ethical relativism is practically ethical nihilism. If one accepted ethical relativism, the holocaust was, from the Nazi's perspective, right. It is just that today we have a different set of values from the Nazis. As the Nazi example demonstrates, we have reason to be profoundly uncomfortable with ethical relativism, even though it is often considered the politically correct thing to be an ethical relativist.
Part of the force behind respecting conscientious objection is a common commitment to ethical relativism: if that is what someone believes, then they are right to believe it, and that alone makes it a kind of truth.
This is not even coherent, much less accurate; it's at a level of absurdity that reminds me of Savulescu's silly 'conceptual analysis' of coercion. There is nothing in this line of argument that needs to be taken seriously.
(1) The whole point of conscientious objection is that the state does not have moral authority over certain matters and thus in those matters cannot coerce citizens; it requires by its very nature that there be moral principles more fundamental than statute or custom. Conscientious objectors are not conformists going along with "groups, cultures or times"; and they are appealing to principles that they believe they themselves don't have the right to violate. There is nothing remotely relativist about this.
Nor does acceptance of a right of conscientious objection commit one to a claim that the conscientious objector is right to believe as he or she does; it merely commits one to a claim that one is wrong to force them against their will in this context. This is a repeated failure throughout S&S's paper: they fail to recognize that a major question is what the state (and a fortiori institutions and organizations less than the state) has the right to require when the state is held to moral standards in the treatment of people in its jurisdiction.
And, in fact, it is not difficult to find recognition that ethical relativism is very unlikely to be consistent with any kind of conscientious objection. Do Schuklenk and Savulescu not read any of the doctors elaborating their views on the subject, and have they never noticed how often opposition to moral relativism comes up? It is commonplace that if conscientious objection were based on a commitment to moral relativism it would be an "anemic" concept. (That's not strong enough, in fact, since it would thereby involve no right, but the point is that it is generally recognized that moral relativism does not have the features required for the kind of appeal to conscience that conscientious objectors need.)
Even someone like Alberto Giubilini, whom they cite for their claim, is not arguing that it is a general commitment that is "part of the force behind respecting conscientious objection". Giubilini's argument in "The Paradox of Conscientious Objection and the Anemic Concept of Conscience" is that moral relativism would give a way around a particular objection to conscientious objection that he thinks is especially cogent and for which its defenders have no particularly cogent response; he is quite clear that this is an alternative that he is proposing as better than what actual supporters of conscientious objection in fact argue. What is more, the conception of moral relativism he brings in is a different kind of relativism than what Savulescu and Schuklenk are proposing. (Indeed, it has to be a different kind if it is to perform the task Giubilini thinks it can.) Their failure to grasp this is truly baffling.
(2) The appeal to the evil of the Nazis in this context is enough to exasperate the most patient of people. One of the evils of the Nazis was precisely their intolerance of rights of conscience, and everyone who is not an idiot knows this. And thus one of the evils of the Nazi regime was its treatment of conscientious objectors, the 'underminers of morale'; conscientious objectors in Nazi Germany were shoved off, like Ernst Friedrich, to asylums or prisons, or, like August Dickmann, to concentration camps and firing squads, or, like Franz Jägerstätter, to the chopping block. It is precisely this -- and similar cases, like Communist treatment of conscientious objectors as enemies of the state -- that has led in the past half-century or so to human rights organizations vehemently supporting the right to conscientious objection in military matters as a human right. There is no sort of ethical relativism in that field. And the question of whether there is a right of conscientious objection in medical matters is simply a question of how far the right to conscientious objection extends beyond the military field. Thus, for instance, supporters of the right to conscientious objection will point out that doctors are not conscripted, and they are not under military-style obligations, and they are not being called up to serve in a national emergency, and yet the state seems to be arrogating a right to treat them as if they are; that is to say, on an analogy between military service and the medical profession, there is even less of a ground for the state to claim it has the authority to demand service, and yet we recognize that authority as less than absolute in the stronger case, so a fortiori we should recognize it in the medical case. There are many varying arguments of the kind, but the point is that recognizing the moral foundation of the one requires recognizing at least the possibility of an analogous moral foundation for the other.
(3) Refusal to recognize any rights of conscience, besides eliminating a major protection against precisely such atrocities as those of the Nazi regime, punishes people for acting as moral agents; if any rights of conscience stand on moral grounds, some right of conscientious objection, even if very limited, is thereby established (since one cannot be defending rights of conscience if no one has any right to refuse due to objection of conscience at all), and it becomes merely a question of what those limits are. Refusal to recognize any right to conscientious objection would be an insistence on penalizing people for trying to act on moral judgments according to rights of conscience; if any right to conscientious objection stands on moral grounds, then the question simply becomes whether those moral grounds suffice for the medical case, as well. Thus questions of ethical relativism are not remotely in view in the grounding of the right of conscientious objection itself; the only way they could come in is if one held that it was needed to handle an objection that could not otherwise be handled (as Giubilini does; his argument, I think, is wrong, but it is at least not sloppy and absurd as the argument of Schuklenk and Savulescu is, which appears to have arisen by mangling Giubilini's argument out of all recognition). But ethical relativism would also gut every one of these steps. (Hence the broad recognition that it would make conscientious objection 'anemic'.)
One of the failings of the argument throughout the paper is that Savulescu and Schuklenk repeatedly fail to recognize that conscientious objection is not opposed to professional ethics but a form of it. The conscientious objector may oppose a law, or may oppose a guideline by an organization, but the whole point of conscientious objection in such cases is that the morality of acting in a profession is not a mere matter of what a law or a guideline says but of more fundamental moral principles. Insofar as we are talking about a profession in particular, conscientious objectors, qua conscientious, are not putting themselves forward as inconsistent with their profession but as opposing something they regard as inconsistent with the objective moral principles governing that profession. Bringing in ethical relativism in the way they do establishes very clearly that they do not have the slightest understanding of what is actually involved in real conscientious objection.