Aquinas's actual discussion in the Summa Theologiae is in the Supplement, which means we don't have his polished view, only the selection drawn from the earlier Sentences commentary. He makes two distinctions -- between sacraments and gifts of God, and between the validity and the lawfulness of the sacrament. The first distinction is to take into account prophetesses. Being a prophet is a higher calling than being a priest, and women can certainly be prophets, so (the objector argues) a fortiori there's no problem with women being priests. And Aquinas concedes that women can be prophets and appears also to concede that prophecy is a higher calling than priesthood. But prophecy is not a sacrament, and therefore is not governed by sacramental symbolism; it is a gift of God given to whomsoever he pleases. "And," says Aquinas, "since in matters pertaining to the soul woman does not differ from man as to the thing (for sometimes a woman is found to be better than many men as regards the soul), it follows that she can receive the gift of prophecy and the like."
So what is the reason given for the inability of women to received the character of ordination? Here Aquinas makes a distinction between validity and lawfulness. If the conditions for validity are missing, you can't receive the sacrament or what it signifies. If I understand the distinction properly, when the conditions of lawfulness are missing, you can receive the sacrament but not what it signifies. So validity has to do with what's strictly required, and lawfulness has to do with what's appropriate, given what the sacrament signifies. Validity has to do with the very possibility of the sacrament, while lawfulness has to do with its failure in particular cases. Or something along those lines. He argues that being a women is an impediment to Orders in both these ways, but his argument is a bit vague:
Accordingly, since it is not possible in the female sex to signify eminence of degree, for a woman is in the state of subjection, it follows that she cannot receive the sacrament of Orders.
And that's the whole argument. In the case of lawfulness I think I can see where he is going. He argues in the case of those in a state of servitude that being in such a state is an impediment to being ordained because it is only fitting (lawful) for those who have disposal of themselves. Where women don't have that sort of freedom, they aren't in a position to be be signs of the appropriate sort, because freedom is required for the sacramental work involved in ordination. Such is more or less the idea, anyway. Validity is trickier, and his position on it depends crucially on a false premise. He distinguishes the two cases by saying that while servants are subject by convention, women are subject by nature, and this is what makes the alleged difference (a slave can validly receive orders, just not lawfully; a woman can receive orders neither validly nor lawfully); and it's in this sense his Aristotelianism makes him take a wrong turn.
It's worthwhile, though, to try to be a bit precise about where the wrong turn comes, because it shows we're not actually so far away from this position as we think we are. Aquinas has argued that, because they have the vital operation of reason, it is fitting for us to be sexually divided as a species; woman is a helpmeet to man in the sense that she is a partner in the work of generation. The 'natural subjection' of women comes as a part of the general need for society to have some sort of governance; this need pre-exists any introduction of sin, and has to do simply with the diversity a good society has to keep in orderly form. So originally people wouldn't have been equal; for instance, some people would have been older and more experienced than other people, and in a good society this would be (at least some) reason for inequality and subjection. But the sort of subjection involved would not be the sort that involves one person being for another's use, nor the sort that involves one person being at another's disposal, nor the sort that involves one person being ordered to another. These are all types of subjection that can only follow sin. The sort of subjection that a naturally good society would have would be the subjection of a free subject to a government directed to his or her proper good and to the common good. There is no distinction between men and women on this system except insofar as it is required for the work of generation. Now Aquinas has the (fairly reasonable) idea that the order of society generally requires that there be someone in charge, if for no other reason than to adjudicate disputes or take responsibility for keeping things going right given the diversity of people. He applies this to sexual diversity, holding that there has to be concurrence between male and female in generation. And this is where we would begin to think that his view of sexual relations goes a little wonky; he usually thinks of it along the lines of the union an active principle and a receptive principle. (This is an analogical extrapolation from the notion of composition -- a composition is a unity of act and potency of some sort; it's a very strong unity, and Aquinas thinks this has to be going on if man and woman are to become one.) That males are by nature fitted to be the active principle (even if in particular cases they fail to be so) Aquinas assumes as a general truth; this is usually assumed without argument, but the 'discretion of reason' mentioned in the passage quoted by the Anglican Scotist appears to be the sort of argument he would give -- apparently he thinks men are better suited for figuring out how children are to be raised, because that would be the primary sort of subjection in question. (Aquinas distinguishes the power to admonish from the power to coerce, and is very explicit that admonishment is the only power in domestic matters. So it comes down to the slightly odd position that the husband always has the authority to advise his wife and decide how to handle the kids; and Aquinas thinks it just so happens that, given original sin, things are arranged so that the wife usually has little choice but to follow the advice.) He never really develops this to the point of being clear; although when talking about the resurrection he talks about "vigor of soul and strength of body". Since he very explicitly insists that women will not be naturally subject to men in the world to come, because that will be a world structured on merit rather than natural need, we have here a glimpse of a different sort of order than the one he typically assumes. Indeed, his chief intellectual problem in this regard is that he hardly thinks about women at all; they come up, inevitably, here and there, but he never gives himself much opportunity to sit down and thoroughly think through what he is saying about them in a systematic way. The impression one gets is that he thinks they're equal but not really; that any inequality is of a very general sort that doesn't apply in individual cases, but does; that it is natural, but won't be some day. It raises a lot of questions, more than he answers.
It is interesting, however, that in this inadequately developed, inadequately polished mass, Aquinas explicitly states (1) that women can have prophetic authority; (2) that they can have delegated spiritual authority; and (3) that they can have temporal authority. Aquinas is juggling a lot of different ideas in his discussions -- natural subjection of women to men, the equality of men and women in matters of the soul, the patriarchal family, the character of marriage as friendship, original sin in this world, merit in the world to come, natural tendencies, individual divergences from these tendencies, the image of God that we must all equally have, Paul's attribution of 'the image of God' to man in particular, the symbolic nature of sacramental work, etc. -- and he never spends enough time to give us a clear argument for this sort of question.