'How are you?' and its rivals in the 1600s
One early instance of "How are you?" as a greeting appears in William Killigrew, Selindra: A Tragy-Comedy (1666):
Poll[idor]: [to Ordella] Madam, I was surpriz'd by your approach, which made my tongue appeare lesse ready than my heart to obey you, your commands, Madam, are sufficient to dissolve any vowes of mine, and since you will vouchsafe to hear, I shall relate.
[Phillocles and Selindra come to them.]
Phill[ocles]: How are you Sir? Methinks your Eyes do shew some anguish in your wounds, do we not trouble you?
However, that wording does not seem to have been the most common form of greeting at that time in England. Thomas Herbert, Some Yeares Travels Into Africa & Asia the Great (1638) provides a series of common English introductory greetings in English, with their Persian equivalents, and "How are you?" does not appear. Here is the English version of the dialogue:
A good morrow or God Blesse you Sir.
The like I with you Sir,
Whether doe you goe?
How doe you today?
Well I praise God
Good, I am very glad thereof,
Where have you beene?
Now I am your servant
Welcome, Sir, heartily welcome.
Tell me, how you doe? healthie
Where is your house? at Babylon
Have you a Wife?
Yea truly, fifteene Sir,
and so on.
A similar series of French/English dialogues appears in Mrs. Mirge et Boyer, Nouvelle Grammaire Angloise (supposedly from 1600, but undated in the photocopied book and perhaps as late as the early 1700s), including this exchange (English version only):
Good morrow, Sir.
I am your Servant.
I am yours.
I thank you.
How do you do?
How do you do, this morning?
At your service.
How are you in health?
How is it with you?
Ready to do you service.
And you, Sir, how do you do?
Very well, thank God.
and so on.
In this exchange (or series of possible exchanges), the question
How are you?" is elaborated as "How are you in health?" which might be the full version of the original question that became truncated to "How are you?" sometime later, but I found very little evidence to substantiate that hypothesis. In any event, the more common inquiry in this period seems to have been either "How do you?" (asked twice in Herbert's dialogue) or "How do you do?" (asked twice in Mirge et Boyer's dialogue.
'How are you?' in the first half of the eighteenth century
In the first half of 1700s, "How are you?" seems to have become fairly well established. Two examples come from Jonathan Swift's letters. First, from a letter from Swift to Sheridan (August 3, 1723) in Swift, Miscellanies, volume 10 (1745):
How are you this Moment? Do you love or hate Quilca the most of all Places? Are you in or out of Humour with the World, your Friends, your Wise, and your School? Are the Ladies in Town or in the Country? If I knew, I would write to them, and how are they in Health?
And second, from a letter from Swift to Lady Acheson (April 1, 1732), in Letters Written by Him and Several of His Friends (1768):
Are you not undone for want of Monky? How are you? Does your milk agree with you? We shall see you no more at church until Monky returns.
From a table of words of one syllable in P. Sproson, The Art of Reading: Or, the English Tongue Made Familiar and Easy to the Meanest Capacity (1740):
how are you to day
From The Trial in Ejectment Between Campbell Craig, Lessee of James Annesley, Esq; Plaintiff and the Right Honourable Richard Earl of Anglesey, Defendant (1744):
Q. Who said it was my Lord's [the earl's] Child?
A. The Country had laid it to him. And so, said I, Juggy, how are you? I'm very sick, says she. What ails you? I have got a Child. Whose is it, says I? It is my Lord's, says she. I believe then, says I, my Lord is very kind to you now, Juggy.
And from a 1748 translation of Terrance's Comedies:
Gnat[ho]. Gnatho greets his dearest dear Friend Parmeno with his best Wishes: how are you?
Par[meno]. On my Legs.
Gnat[ho]. Pshaw, I know that : —— but dost thou see Nothing here that thou dost not like ?
Par[meno]. Yes, you.
A translator's note to these lines remarks
Quid agitur sometimes means what are you doing? Sometimes how do you? How are you? or how goes the World with you?
"How are you?" appears in Google Books search results as stand-alone question going back to 1666. At the time "How do you?" may have been considerably more common as a salutation; and "How do you do?" provided further competition later on.
Two of the pre-1750 matches that I found for "How are you?" complete the question with "in health?" so it may be that "How are you in health?" was actually the earlier question, shortened to "How are you?" after the phrase became commonplace. It is not clear to me from the data Google Books provides, however, that this process occurred. The only conclusions I can state with confidence is that "How are you?" has been standing alone in inquiries after to a person's health or well-being for more than 350 years.