Our morality is intrinsic to evolution, I would argue that the more moral a species is on a co-operative level, the more likely it is to triumph against the evolutionary threats to its survival. Thus everything is not permitted, as evolution keeps weeding out the non-co-operatives.Yes, as a social construct and likely from innate response mechanisms.
However, morality as a foundation of being, as something intrinsic to the reality of the Universe, no. Without a God, morality becomes relative and "everything is permitted" as long as consequences are accepted.
To provide a simple example, take child killing. A child killer, quite aside from being assessed as a threat and being removed by the co-operative members of society, is also more likely to remove their offspring from the gene-pool. So in a sense we evolved away from child killing. As I'm sure you will see if you ponder, God is not required to actively weed out child killers. I personally feel that it could be argued that our morality is evolutionary in nature, and therefore exists as independantly as it is possible for anything to exist from God. In other words what i mean to say is that taking God out of the equation does not change our evolutionary history, or our morality that necessarily evolved with it.
By the way I am a monotheist, but only because I am hardwired to believe in God, not because morality is logically dependant on God.