St. Thomas addresses the heresy of Nestorius in SCG, IV, 34. It is a fairly lengthy chapter, and we won't be looking at every single remark that Aquinas makes in it. He starts off by introducing the error:
They said that the human soul and the true human body came together in Christ by a natural union to constitute one man of the same species and nature with other men, and that in this man God dwelt as in His temple, namely, by grace, just as in other holy men. ... So, then, consequently on the things just said there must be one Person of the Word of God, and another person of that man who is co-adored with the Word of God. And if one Person of each of the two be mentioned, this will be by reason of the affective union aforesaid; so that man and the Word of God may be called one Person, as is said of man and woman that “now they are not two, but one flesh” (Mat. 19:6).
Now, such a union does not bring it about that what is said of the first can be said of the second (for not everything which becomes the man is true of the woman, or conversely); therefore in the union of the Word and that man they think this must be observed: The things proper to that man and pertinent to the human nature cannot be said becomingly of God’s Word, or of God. just so it becomes that man that he was born of a virgin, that he suffered, died, was buried, and this kind of thing; and all of these, they assert, ought not be said of God, or of the Word of God. But, since there are certain names which, although they are chiefly befitting to God, are nonetheless communicated to men in a fashion—“christ,” for instance, “lord,” “holy,” and even “son of God”—nothing according to them keeps one from the use of such names in predication of the things just mentioned. For, according to them, we say fittingly that Christ, or the “Lord of glory,” or the “Saint of saints,” or “God’s son” was born of a virgin, suffered, died and was buried. Hence, too, the Blessed Virgin must not be named the mother of God, or of the Word of God, but the mother of Christ, they say [SCG IV, 34, 2 passim; emphasis added].
And here we get to the heart of why it is that sometimes Catholics say that Protestants are Nestorian: they do not think it fitting to use certain sorts of language when speaking about the Lord Jesus Christ. This will be somewhat lengthy, but I think it's worth examining a bit more. The point isn't, as I already said, that Protestants are necessarily Nestorian; rather, it is that it seems (for some of them at least) their use of language, and their scruples about it, undermine the Incarnation after a Nestorian fashion, and that it would be better for them to modify that usage to be more orthodox.
if “the Word was made flesh,” that is, “man,” as the Evangelist witnesses (John 1:14). it is impossible that there be two persons, or hypostases, or supposits of the Word of God and of that man [ibid., 5; hereafter references to this same chapter will be by paragraph number only].Now that may seem pretty obvious to us today. But St. Thomas draws the conclusion, "Whatever was made is what it was made; thus, what was made man is man, and what was made white is white. But God’s Word was made man, as is gathered from the foregoing. So God’s Word is man" [ibid.; emphasis added]. And:
Demonstrative pronouns, moreover, refer to the person, or hypostasis, or supposit. For no one says “I run” when another is running, except figuratively, perhaps, when another is running in his place. But the man called Jesus says about Himself: “Before Abraham was made, I am”,and “I and the Father are one” (John 8:59; 10:30), and several other things which clearly pertain to the divinity of the Word. Therefore, the person and hypostasis of the man speaking is plainly the very person of the Word of God .The point here is that the second person of the Trinity is identified with the man Jesus - they are the same person. And of course the average orthodox Protestant (by which I mean one who accepts orthodox Christology) is not going to argue with that.
This usage appears frequently in Scripture.
- Jesus says he is the bread which came down from heaven, which could only be said of the Word (John 6:51) 
- The disciples watched him ascend to heaven (Acts 1:9), but St. Paul says "He that descended is the same also that ascended above all the heavens" (Eph. 4:10); the disciples could not see Christ's divinity ascend into heaven, but it is only his divinity that could have descended from heaven 
- Jesus says that he came into the world (John 16:28), but that which has its origin in the world (namely, his body) cannot properly be said to "come into" it; so it must be the Word of God who speaks 
- The suffering of one's body can be ascribed to the one whose body it is, and consequently we may say that the Word of God suffered, was crucified, died, and was buried because these things happened to his body 
- The psalmist says that God is the King of glory (23:8-10), and Paul says that Christ is the Lord of glory (1Co 2:8); hence we may say that God was crucified 
- God delivered his own Son up for us (Rom. 8:32), so that it is right to say that the Word of God suffered and died for us 
- "[O]ne is said to be the son of a mother because his body is taken from her, although his soul is not taken from her, but has an exterior source. ... Now, it was proved that the body of that man is the body of the natural Son of God, that is, of the Word of God. So it becomes us to say that the Blessed Virgin is “the Mother of the Word of God,” and even “of God”. Of course, the divinity of the Word is not taken from His Mother, for a son need not take the whole of his substance from his mother, but his body only" 
- "And, again, John says: “The Word was made flesh.” But He has no flesh, except from a woman. The Word, then, is made of a woman; that is, of the Virgin Mother. Therefore, the Virgin is the Mother of God the Word" 
- Paul says that Christ is "of the Fathers" (Rom. 9:5), but this is only through his Virgin Mother; therefore he who is "over all things, God blessed forever" (Rom. 9:5 again) is of the Fathers through the Virgin Mother; therefore she is the mother of God in the flesh 
- St. Peter says that the man Jesus has been exalted to the right hand of God (Acts 2:33), and St. Paul says that God "emptied himself" (Php. 2:7); hence, if we may say that the man Christ Jesus has been exalted, we may also say that "lowly thing" may be said of God - that he suffered and died, or that a Virgin was his Mother 
- St. Paul says of this one person that in and by him all things were created (which could only be said of him in his divine nature), and that he is the firstborn from the dead (which could only be said of him in his human nature); hence "whatever is said of that man must be said of the Word of God, and conversely" (cf. Col. 1:16-18) 
Now this is something to which at least some Protestants object. While it is undoubtedly true that at least some of these objectors hold an orthodox Christology, nevertheless their unwillingness to accept perfectly appropriate language concerning Christ and his Mother sounds as if they deny the orthodox doctrine of the Incarnation, by denying the unity of the Person of Christ. He is one Person, of whom we may say that he is God, and that Mary is his Mother, and that he suffered and died for us, and that in him all things were made.
[Update - to add some additional considerations here...]
Why does it seem at face value as though they deny the orthodox doctrine? Because to be unwilling to say that Mary is the Mother of God - to be only willing to say that she is the Mother of Christ - is to suggest that the child - the person in her womb was not the Word of God. Now that may not be what they intend, but it is nevertheless the implication that is hard to avoid. But if he was not the Word, then he was merely human. And that, of course, would be heretical.
This is why it is so important to be willing to say that Mary is the Mother of God - the theotokos, the God-bearer. And as we have seen above from what St. Thomas has argued, we have excellent precedent for saying this, inasmuch as the Scripture often says of the human nature what rightly applies only to the divine, and vice versa. One who gives birth to a child is that child's mother; but Mary's child was God. Hence Mary is the Mother of God. Protestants who are distressed by this should carefully reconsider the implications of a denial of it.