It ultimately matters what kind of God is believed in. God isn’t a piece of furniture that can be wheeled in to hang the “important morality stuff” on. I know C. S. Lewis attempted to make a case by consensus for a “Tao of man” but I think even he failed to note (maybe he honestly didn’t know) yes, there is at least one known place on earth where men are proud of double crossing those who had been kindest to them. And the like, which he likened to a place where 2 + 2 was believed to equal 5.
Of course it belief in God as He is matters.
But belief in God is voluntary, whereas standard morality upheld in society is. I don’t mean criminalizing adultery or stoning adulterers; but some things that are legal now should not be (pornography, sodomy) and some things that are “celebrated” or tolerated should be severely frowned upon and given some legal weight.
Belief in God, specifically particular denominations or sects or religions, is not something that the government should mandate or favor, but the universal morality that is taught in all religions should be.
BTW the law of karma can be stated in non-Sanskrit words:
As you sow, so shall you reap.
Pretty simple.