Does our ability to act morally depend on who we worship?
Yes. If we don't value God for who he really is then our behavior, which is intended to be the fruit of our valuing of God, is going to reflect that skewed understanding of God.
The very essence of morality is not the deed that we're doing—such as helping somebody change a tire on a cold winter night or not stealing—but the essence of morality is the mindset out of which the deed is growing. It is the deed together with the mindset.
If the mindset has its roots in a flawed perception of God then the God who is being reflected through the deed is going to be poorly reflected. I believe the reason we tend to think that morality is not affected by a flawed view of God is because we don't understand the essence of morality as being the mindset, the motive, and the display of God.
This is why I have a problem with talking about a "Judeo-Christian ethic". If you say "Judeo" and you mean Jews who do not believe in Jesus Christ but hold to the Ten Commandments, then you're introducing a flaw into worship which is utterly profound.
The New Testament is written to say that those who do not honor the Son do not honor the Father. So the concept of a Judeo-Christian ethic as the goal to which people ought to aim is profoundly mistaken, because ethics has to grow out of a true view of God. And to reject Jesus Christ is to have an absolutely flawed view of God. Therefore the ethic of morality that flows from this kind of flawed view of God is going to be flawed, even if some of the behavior is the same.
The point of ethics is not merely the shell of the behavior, but it is the inner convictions of the mind, the disposition of the heart, and the goal of what we're displaying. If Jesus Christ is omitted from that then I don't think we have Christian ethics or morality.