Has the distinction of the human and divine natures in Jesus Christ been established?

On the contrary, mainstream Christianity has held that the human and divine natures of Christ are not separable. A key point was the third ecumenical council, held at Ephesus in 431 in order to resolve disagreements between Nestorius, the Patriarch of Constantinople, and Cyril, the Patriarch of Alexandria. Nestorius wanted to make a distinction between Christ the person, and the eternal Son: two distinct natures in one body.

Cyril's anathemata against Nestorius, which the council accepted, express the view that it is impossible to draw a sharp line between the human and divine natures. It's worth unpacking a few of these:

(2.) If anyone shall not confess that the Word of God the Father is united hypostatically to flesh, and that with that flesh of his own, he is one only Christ both God and man at the same time: let him be anathema.

This is an expounding of the hypostatic union idea, whereby the human and divine essences are joined, making Christ simultaneously fully human and fully divine.

(3.) If anyone shall after the union divide the hypostases in the one Christ, joining them by that connection alone, which happens according to worthiness, or even authority and power, and not rather by a coming together, which is made by natural union: let him be anathema.

The union is a "natural union", not an imposition of the divine over the human. This counters the Nestorian position that Jesus was merely the carrier for the Word (the human nature being subordinate).

(4.) If anyone shall divide between two persons or subsistences those expressions which are contained in the Evangelical and Apostolical writings, or which have been said concerning Christ by the Saints, or by himself, and shall apply some to him as to a man separate from the Word of God, and shall apply others to the only Word of God the Father, on the ground that they are fit to be applied to God: let him be anathema.

So Cyril says that in talking about Christ, it's wrong to apply separate terminology or titles for the human and divine natures. Again, this counters the idea that some of Christ's actions can be attributed to Jesus the human (such as his eating, weeping, dying, etc.) and others to the Word (responsible for our salvation, all things were made through him, etc.).

Twenty years later, the Council of Chalcedon of 451 adopted a creed that expresses the relationship more briefly:

We, then, following the holy Fathers, all with one consent, teach men to confess one and the same Son, our Lord Jesus Christ, the same perfect in Godhead and also perfect in manhood; truly God and truly man, of a rational soul and body; consubstantial with the Father according to the Godhead, and consubstantial with us according to the Manhood; in all things like unto us, without sin; begotten before all ages of the Father according to the Godhead, and in these latter days, for us and for our salvation, born of the Virgin Mary, the Mother of God, according to the Manhood; one and the same Christ, Son, Lord, Only-begotten, to be acknowledged in two natures, inconfusedly, unchangeably, indivisibly, inseparably; the distinction of natures being by no means taken away by the union, but rather the property of each nature being preserved, and concurring in one Person and one Subsistence, not parted or divided into two persons, but one and the same Son, and only begotten, God the Word, the Lord Jesus Christ, as the prophets from the beginning have declared concerning him, and the Lord Jesus Christ himself has taught us, and the Creed of the holy Fathers has handed down to us.

The two natures are distinct insofar as they are not the same (humanity and divinity are different things), but they are united inseparably in Christ.