Deep down, you know that's not true.
I know no such thing, and please don't presume to tell me what I know and don't know.
If there is an objective standard of morality, a moral law if you will, that could only mean a moral law giver.
No, that conclusion doesn't necessarily follow, but even if it were correct, it also doesn't follow that the law giver cannot be humans. There are things that are universally true about human nature, we are social animals with the same needs for love and growth, we respond similarly to similar situations, and so forth. It can hardly be a surprise that we shape our rules and institutions to meet common human needs. Those common human needs provide a perfectly satisfactory explanation for human morality.
..if the moral standard they appealed to doesn't exist, and therefore morality is relative...
You're not thinking clearly. If the moral standard they appealed to doesn't exist, then morality is neither absolute nor relative, it's not there at all. Just because it's relative doesn't mean it doesn't exist.
