If I understand the White House announcement correctly, their newly proposed rule would not actually change the moral circumstances at issue in any way.
The problem that opponents of the original rule have had is that it effectively requires religious employers to purchase a product (an insurance policy) that provides their employees with free access to contraceptive and abortifacient drugs that they would not have otherwise had, and thus requires those employers to purchase a product that violates their religious convictions. The new rule does exactly the same thing.
It puts religious employers in the position of having to choose between providing their workers with free (to the workers) access to contraceptives and abortifacient drugs or not providing those workers with health insurance at all (and also paying a large fine). The only difference is that the access to those contraceptive and abortifacient drugs would not technically be listed as one of the benefits the employer was paying for directly but would be listed as a benefit the insurer was paying for (with the money the employer paid for the broader insurance policy, of course). But employers who offer insurance don’t pay for individual benefits and products when they are provided anyway, they pay for the policy that gives their workers access to those benefits and products when they want them. Under this rule, then, it would still be the case that as a result of being employed by a religious institution that provides insurance coverage (which Obamacare would require employers to do, or else pay a large fine), workers at that institution would have free access to contraceptives and abortifacients that they would not have had if that employer did not offer insurance coverage. So it’s still the case that the rule would require religious employers to purchase a product that violates their convictions, in the same way as the original rule (a fact also highlighted by the administration’s decision to retain the exemption for actual houses of worship in this new rule, just as in the old one). The choice for religious employers is still between paying an insurer to provide their workers with access to a product that violates their convictions or paying a fine to the government.