What is the ruling on saying "By God, by God, by God, if you do such and such, then you are on one path and I am on another path," with the intention of swearing and out of anger, and does it necessitate expiation for breaking an oath?

The preponderant view is that oaths are based on intention. So, if your intention by saying to your wife, "You are on one path and I am on another," was that you would become angry with her if she did a certain thing, then if she did that and you became angry with her, your oath is fulfilled and no expiation is required of you. The same applies if the wife did not do that thing at all. However, if she did that and you did not become angry with her with the intended anger, then you have broken your oath and an expiation for an oath is required of you.