Understanding moral laws as divine commands is not by itself to have a religious morality. To be sure, understanding moral laws as divine commands involves a natural piety towards morality. But someone need not belong to any religion to understand moral laws as divine commands. This is compatible with holding that understanding moral laws as divine commands is more than Moral Deism .
Divine command moral theory makes indicative claims about human nature that are in a way falsified by natural science. It claims that there are ends in nature which ought never be frustrated. This claim is falsified by science in the sense that it is an inadmissible scientific statement. It cannot be true if scientism is true.
The title is Divine Command Morality and Religious Morality. But the more accurate title would be Divine Command Morality and Catholic Christian Morality. I do not know enough about the code of the vast variety of religious to compare religious morality in general with plain morality. My paradigm of religious morality are prescriptions of Jesus in Matthew’s, Ch. 5-7, account of the “sermon on the mount” in which Jesus says: ” You have heard it said but I say to you .. .” For instance:
38 Ye have heard that it hath been said, An eye for an eye, and a tooth for a tooth:
39 But I say unto you, That ye resist not evil: but whosoever shall smite thee on thy right cheek, turn to him the other also.
The code of Catholic morality contains all of morality plus other norms of two types. There are ritual prescriptions and religious moral commands. I do not want to digress into trying to define ritual prescriptions. Suffice it to say that they are not regarded as for anyone outside the religion. The religious moral commands are regarded as applicable for all human beings. How do religious commands differ from plain moral commands? . They lack the necessity of moral commands. We can think of them not being given. For instance, we cannot think of adultery being morally permissible. However, we can think of remarriage after one partner has abandoned his spouse being permissible. Jesus is quite explicit that he is adding to what has been morally taught with “you have heard it said, but I say to you.”