I don't even understand the objection. Are you saying that 'the left' are proving that they have objective morality by hating America? I don't see how the two things are related.
And anyway, no one has objective morality. Claiming "God says it's wrong, so it's wrong" is just as subjective as claiming "I believe it is wrong, and so it's wrong." I believe that what my fellow Christians usually mean when they say that atheists lack objective morality is that, in comparison to those who believe in God, Whose thoughts and ways are not like ours (i.e., Who exists outside of mankind's shifting notions of morality, as the eternal "Just Judge"), atheists lack a belief in the concept of an eternal, unchanging standard of what is right and wrong. Whether or not you need to believe in God in order to have such a standard is above my paygrade to speculate on, but I could see how a religious person like your average western Christian (or even just a person raised in a western/post-Christian society) could come away with that impression, given a certain way of looking at history and human motivation absent some kind of rightly-feared corrective force or thing (which is what God is to a lot of religious people).