In one of Newton's manuscripts (Yahuda Ms. 1.4) we find a discussion of the Arian controversy in the context of an exegesis of a vision in the book of Revelation, with which he had a bit of a fascination. On Newton's view, the primary purpose of Revelation was "to describe & obviate the great Apostacy," which "was to begin by corrupting the truth about the relation of the Son to the Father in putting them equal." For this reason, on Newton's view, the vision in Revelation 5, in which a figure is seen upon a throne and gives to the Lamb a scroll, is not a mere set of images by a doctrinal system put in imagistic form. In particular, it is a prophecy showing the true relation between Father and Son: "the Son's subordination, & that by an essentiall character, his having the knowledge of futurities only so far as the father communicates it to him." The scroll is originally sealed; Newton points to this as evidence that the Son's receiving of knowledge from the Father is not eternal. This knowledge "was not given to the Lamb at his first generation but since his resurrection; he meriting it by his obedience to death."
The obvious orthodox response to this, of course, would be to say that Newton is confusing the Word in Himself with the Word Incarnate. Newton, however, is no fool, and anticipates the response, which he thinks the vision also guards against "by a threefold insinuation."
First, the vision begins with the one on the throne holding the book in his hand, and is closely followed by the declaration, in the entire company of heaven, that only the Lamb is worthy of it. Thus the vision shows God and the Lamb as the most worthy in this assembly; and the Lamb is shown originally without the book. On Newton's view, this suggests that the distinction between the Word as God and the Word as Incarnate is sophistical: if the Word had known these matters beforehand, the Lamb would have been as much in possession of the book as the one on the throne. (For those who aren't used to this sort of topic, Newton's argument here is very weak; it depends crucially on his assumption that the scroll is some sort of divine knowledge. The usual interpretation, which takes into account the whole imagery of the book of Revelation, is that the scroll has to do not with knowledge but with salvation and judgment. The one thing going for Newton's interpretation is the claim in the Gospels that only the Father, and not even the Son, knows the day and hour of judgment; this verse is an important problem for the orthodox position, but Newton's application of the claim to the vision is a stretch.)
Second, Newton notes that the Lamb in this vision is the object of worship, both alone and together with the one on the throne. This might at first seem to cause a problem for Newton's own view, but he has a clever response that is relevant to his jab at Trinitarianism in the General Scholium:
Now this worship was given to the Lamb as he was a God without all doubt, Divinity & worship being relative terms, & yet it was given to him as he was worthy to take & open the Book for at the falling down of the four Beasts & 24 Elders before him to worship him, the very act of their worship was to celebrate him for his worthines to take & open the book. The Lamb therefore as he was a God was worshipped for his worthines to take & open the book & therefore took & opened the Book as he was the object of worship, that is a God. But to make all this plainer you may compare it with Philip: 2.9 where tis expresly said, that for his obedience to death God gave him a name above every name that at the name of Iesus every knee should bow &c. that is that all the creation should worship him which is as much as to say that he should be ισα θεω as a God over the creation: for Deity & worship are relative terms & infer one another.
In other words: 'God' is not an absolute term; it doesn't identify anything ontological. The reason we call something a 'God' has nothing to do with what it is in itself. Rather, we call a thing 'God' if it has the sort of dominion or authority that calls for worship. Thus Newton has no problem with calling the Lamb 'God', because the Lamb is given divine authority (which he previously did not have) by the Supreme God (the one on the throne).
Third, Newton identifies a difference in how God and the Lamb are treated by the vision as objects of worship. (1) The Lamb does not sit on the Throne but stands by it; whereas the one on the throne (and who is therefore King over all who are not on the throne, including the Lamb) represents God. (2) In Newton's view, the doxologies that follow the investiture of the Lamb show a gradation, with God being given a "higher degree of worship" than the Lamb, a pattern that he thinks is repeated in Revelation 7.
Thus Revelation 5 is:
a system of the Christian religion, showing the relation of the ffather & Son, & how they are to be worshipped in a general Assembly of the Church & of the whole creation. The ffather the supreme King upon the Throne, the fountain of prescience & of all perfections. The Lamb the next in dignity, the only being worthy to receive full communications at the hand of the ffather. No Holy Ghost, no Angels, no Saints worshipped here: none worshipped but God & the Lamb, & these worshipped by all the rest. None but God upon the Throne worshipped with the supreme worship; none with any other degree of worship but th eLamb; & he worshipped not on the account of what he had by nature, but as he was slain, as he became thereby worthy to be exalted & indowed with perfections by the father. This was the religion to be corrupted by the Apostacy. This therefore was very pertinently shaddowed out in the exordium to the Prophesy of that Apostacy.
This is Newton's Arianism in a nutshell; although I think it's a bit strained, it's quite striking, and much more creative and original than most subordinationisms.
On to the Principia. In the General Scholium, which was added to the Mathematical Principles in 1713, after having stated that a system as beautiful as the solar system must be "under the dominion of One," goes on to consider the nature of this being:
This Being governs all things, not as the soul of the world, but as Lord over all: And on account of his dominion he is wont to be called Lord God Pantokrator, or Universal Ruler. For God is a relative word, and has a respect to servants; and Deity is the dominion of God, not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: These are titles which have no respect to servants. The word God usually a signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God; a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God.
It can be seen easily enough that this is the same argument that we saw above. There it was used to interpret the prophecy of Revelation in a non-Trinitarian way. Notice the claim that "a being, however perfect, without dominion, cannot be said to be Lord God". (As we saw above, Newton interprets claims that the Son was given dominion quite strictly with regard to the person of the Son.) Notice also the distinction between 'true', 'supreme', and 'imaginary' Gods. (As we saw above, according to this distinction the Word is a true God, but not the Supreme God.) By relativizing the term 'God' in this way, Newton can break up the apparent unity that seems to be attributed to Father and Son in Scripture. Of course, by relativizing the term 'God' in this way, Newton seems to be committing himself to polytheism. He does, however, make some effort to alleviate this by pointing out occasional uses in Hebrew and the like where the relevant term for 'God' is applied to people who aren't God. This is fair enough. It's doubtful, however, that this bit of evidence stretches quite as far as Newton wants it to stretch.
ADDED LATER: A number of people have been coming to this recently for various reasons, so I thought I would update it with a link to the newer address of the Newton Project, where you can find the manuscripts in question: