Something many theists point to is without God, morality is subjective. For example, an atheist can't justify that murder is intrinsically wrong. If God does exist however, then morality is objective and murder is intrinsically wrong. On the surface this makes sense, but why is this actually the case. A government or authority figure can't justify intrinsic moral values, but if God exists then he can? Even if God does exist, there is still opinion on whether we should listen to him or not (e.g. Lucifer's position).
If we say objective morality exists, we surely mean by it that some things are moral or immoral (good or bad) independent of any other variables. That would include time and space. Which would make objective morality either god, or proceeding in eternity past from God.
A government or authority cannot produce objective morals because it cannot exist outside any variables. It can recognise it though, but may not have a basis to appeal to it (justify it) as being objective unless it can appeal to the Divine. Which is what the Declaration of Independence does to imbue certain unalienable rights.
The theist's argument is that the recognition of the existence of objective morality is evidence that something preceded the existence of variables.
This would imply if man can recognise objective morality, that the Source of this morality has revealed it or "coded" it into creation.
The Bible teaches that objective morality (the righteousness of God) is naturally revealed to man (Ecc 7:29, Rom 1:19) but was expressly revealed in God's Law and Prophets, but ultimately manifest in the person of Christ (Romans 3:21-22).
You posed a good question. You are right in that God's existence alone is insufficient to guarrantee the objective basis of morality. Since this is a Christian site, I will outline how objective morality flows from a Christian's understanding of God and human nature.
You are also right that a government or an authority figure alone can't justify intrinsic moral values. However, if the constitution is based on something objective such as natural law theory (like in the United States), then there is some measure of objectivity. See this 2015 paper from the Journal of Constitutional Law: Man, Morality, and the United States Constitution. Some quotes (emphasis mine):
A. Natural Law Theory
Although natural law theory has ancient and medieval roots, the strand that most influenced the American Framers developed and matured in the time directly preceding the Enlightenment and its early stages. The seventeenth century is often referred to as “the century of genius” primarily for the large change in gestalt that occurred. While the late seventeenth century is often pegged as the starting point of the Enlightenment, the work done by Francis Bacon, Isaac Newton, John Locke, as well as the natural lawyers in the early part of the century, paved the way for the latter movement. These thinkers forever changed how people view phenomena, shifting explanation away from the use of tradition and towards the use of reason. Early Enlightenment thinkers were motivated by a “systematic spirit.” No endeavor, be it practical or theoretical, escaped their quest to produce rational and systematic explanations of the world around them. The natural law theorists, Hugo Grotius, Samuel Pufendorf, and John Locke, extended this quest to morality. They sought to understand morality and God’s role in moral laws in a legalistic way consistent with new, Protestant ideas about God and his role in the affairs of man. Furthermore, they sought a systematic account of how positive law, the laws of men, fit with the laws of morality.
1. Reason and Moral Epistemology
The natural law theorists of the seventeenth century placed a strong importance on human reason and the ability to figure out moral truths. This shift toward reason was the result of the shift away from a god who was directly involved in human and earthly affairs. The Protestant God was a god who was with disconnected from man. The link between man and God was through human reason. It was through reason that man could ascertain the right and wrong actions to take. There were two different views on the method of reasoning to arrive at God’s law. Grotius favored an inductive method to arrive at natural law. On the Grotian account, in order to arrive at the natural laws an examiner needed to explore the positive laws and customs of various countries. Later successors, Pufendorf and Locke, would reject this inductive methodology and seek to explain knowledge of natural law and moral principles through deductive reasoning.
As you can see from the quote above, there was an objective basis for morality in the design of the United States constitution, but because the US constitution was born in the spirit of the Enlightenment philosophy the medieval link between reason and God was severed, and as a result Enlightenment era natural law understanding was also severed from God and His Ten Commandments. In the 21st century we are now far removed from that Enlightenment confidence in the objectivity of reason without God. As a result, in today's climate we see a steady erosion of objectivity in public morality, making moral truths relegated to what the powerful say in the courts, in Congress, and in the justice system (executive branch).
BUT for Christians who believe that natural law is discoverable by reason precisely because God puts it within human nature, even in the post modern age we are in today, we can still use reason to discover objective moral truths like "do not murder" since we literally carry the law within the fabric of our being (instead of being instilled by society). This discoverable natural law is consistent with the divine laws that God revealed to us in the Bible (such as the Ten Commandments) since Christians of all stripes believe that the author of both is the same.
A government or authority figure can't justify intrinsic moral values, but if God exists then he can? Even if God does exist, there is still opinion on whether we should listen to him or not (e.g. Lucifer's position).
Christians believe that these moral values are intrinsic to us because God instills them within our human nature, hence the name "natural law". It is objective because Christians believe human beings share the same nature, which is based on the teaching that we are made in the image of God. Part of this human nature is free will that enables us to either obey or disobey this natural law. Another element of this human nature is conscience who nudges us to obey but whose voice we can disobey because of free will. The Bible is replete with verses that say how human beings can flourish if we obey. So obviously God wants us to obey.
Should we obey God? That is a personal decision and God will demand accountability from each of us at the last judgment.