Dov Fox has a paper, What Will Happen If Doctors Defy the Law to Provide Abortions? (PDF), in which the argument is made for conscience protections for abortion providers. I suspect similar arguments will become more common; under Roe, the common argument was that there should be no conscience protections at all in questions of abortion, and now that Roe is gone, people who would previously have argued in this way are hastily trying to build a conscience-protection shelter for their preferred positions. We should be glad, nonetheless, that there is a renewed recognition of the importance of conscience in medicine. It's certainly less morally and intellectually defective than the older line of argument, but the kind of argument given in Fox's paper is still difficult to take seriously when the argument makes so many elementary mistakes about conscientious objection.
(1) The argument confuses conscientious objection and civil disobedience. While one can appeal to conscience in the latter as well as the former, it is structurally different. Conscientious objection is something that can be potentially incorporated into laws and policies, and almost always concerns omissions; civil disobedience is by definition inconsistent with something recognized as law, and usually concerns commissions of acts that are illegal in that sense. Because of that, they require different justifications -- conscientious objectors are only asking to be accommodated, while not thereby interfering with the legal policy, and therefore their request is grounded on personal considerations; those engaged in civil disobedience are treating a law as immoral either in itself or in its application, thus implying that the law should be eliminated for everyone, and therefore their request has to be grounded on some sort of higher law or moral principle. Due to these differences, they are also not addressed in the same way.
(2) A key problem with Fox's argument is that the conscientious objection side is framed in terms of "denial of care". But conscientious objectors are not denying care, any more than someone denies care if they judge that they do not have the skill or the resources to engage in a given medical intervention properly; the only difference, in fact, is that in this case they are saying that they are morally, rather than technically or materially, not the right sort of person to provide a given medical operation. The notion that doctors' moral judgments are just some extra and irrelevant thing with no business in medicine is absurd; doctors have to assess whether they can properly provide any proposed medical service on material, technical, legal, and moral grounds, and if they deem that they are not able to do so, on any of those grounds, they have a moral responsibility not to do it.
(3) Fox says, "Forcing doctors and nurses to stand by and do nothing to help patients in need flies in the face of clinicians’ fundamental charge to heal, promote health and relieve suffering"; but it's almost universally recognized that the fundamental charge of clinicians is to do no harm, and positive charges are secondary to this. The symmetry that Fox sees between 'conscientious providers' and 'conscientious refusers' does not exist, even in other areas. If hospital policy restricts provision of a certain drug to those who have gone through a certain process, and a doctor provides the drug without the process, on the grounds that it relieves suffering, the doctor will have usually acted unprofessionally and in violation of basic responsibilities, by short-circuiting a process intended to protect patients; mere appeal to relief of suffering is not ever adequate to justify any medical treatment or intervention. But the inappropriateness of a medical treatment or intervention is usually adequate to justify not doing it. Those who take the totalitarian position that medical conscientious objection is not allowed, will not accept cases where the judgment is that the treatment or intervention is morally inappropriate, but Fox in principle accepts that a judgment of moral inappropriateness can be considered. That's all that's required for the asymmetry, though.
(4) Not all of Fox's arguments are bad. He rightly notes the importance of the patient's wishes, as well as the importance of giving a fair amount of room for conscience in a pluralistic society. These are both legitimate considerations. They do not, however, give us much guidance on this issue, precisely because of the asymmetry: patient's wishes and clinician's conscience should both weigh heavily on what should not be done; they have to be joined with a lot of other considerations to decide what should be done. (It would be legitimate to argue, however, that both patients' wishes and clinicians' consciences should at least be given consideration in forming any law or policy on a matter; these are indeed the kinds of things that legislators and policy-makers should often consider.)