Learn English – greet with/through eyes

Using via as a preposition in English is of comparatively recent provenance. It has substantially fewer primary senses, and therefore available uses, than does through. The OED gives only two main senses for via as a preposition, which I include here with a few of each one’s later citations:

1. By way of; by the route which passes through or over (a specified place).
   
   
   • 1958 A. Sillitoe Saturday Night & Sunday Morning iv. 60 — Arthur and his father walked via the scullery into the living-room.
   • 1959 M. Gilbert Blood & Judgement xiii. 138 — More··had come to the Police via the Lower Deck of the Royal Navy.
   • 1981 G. Household Summon Bright Water iii. 149 — He led me to talk of my interest in ancient economies and thus, via agriculture in the Forest of Dean, eased the way to my impressions of Broom Lodge.
2. By means of, with the aid of.
   
   
   • 1972 M. Kaye Lively Game of Death (1974) vii. 41 — Any deal··would have to be··concluded via contracts, attorneys, the whole schmeer.
   • 1977 Rep. Comm. Future of Broadcasting iv. 30 — It would in theory be possible to provide five more services with national coverage via satellite.

Here are some examples of through (taken from the OED’s citation list for that preposition) where you could not substitute in via in its stead:

• 1847 Tennyson Princess iv. 554 - Thy voice is heard thro’ rolling drums.
• 1848 Thackeray Van. Fair xxxii, — George··was lying··dead, with a bullet through his heart.
• 1852 R. S. Surtees Sponge’s Sp. Tour (1893) 85 — He was small and wiry, with legs that a pig could run through.
• 1886 Ad. Sergeant No Saint I. vi. 105 — An old land surveyor··put him through a long catechism.
• 1896 T. F. Tout Edw. I, iv. 80 — All through his reign, the Lusignans helped him in Gascony.
• 1903 Times 14 Mar. 14/5 — The Oxonians showed good form through choppy water.
• 1975 Nature 10 Apr. 501/2 — Nine recognised glaze types, ranging in colour from pale blue, through green, to yellow, brown and red.
• 1981 L. Deighton XPD xliii. 342 - A··notice stating that deliveries were only accepted between eight and eleven Monday through Friday.
• Mod. — There is a path through the wood.
• Mod. — It has passed through many hands since then.

In contrast, in these examples from the same source, one perhaps might be able to make that swap:

• 1852 Dickens Bleak Ho. viii, — Mrs. Pardiggle··had been regarding him through her spectacles.
• 1885 Act 48 & 49 Vict. c. 53 §15 — Every notice··sent through the post in a prepaid registered letter.
• 1894 J. J. Fowler Adamnan Introd. 56 — The southern Picts··embraced the truth through the preaching of St. Ninian.

So even though though there are a few places where you can use via or through — or else via and by — interchangeably as prepositions, there are many others where you cannot.

Finally, it should be noted that there are substantive, adjectival, and adverbial uses of both words, and that these non-prepositional uses are never interchangeable.

I believe this to be a case of inversion of an accepted alternate form. In other words: many times questions are formed by inversions of statements and this is an inversion.

What color[ed] eyes did she have?

Inverts (more or less):

She had blue colored eyes.

There is a tendency in spoken language to swallow the -ed at the end of a word. And, this tends to find its way into the written form. Hence color vs. colored.

When you consider this a truncation of the -ed, you will see that a preposition is no longer necessary because it is an adjective being used as an adverb.

Consider another case:

What size shoes do you wear?
I wear size 7 shoes.

Most people don't say sized 7 shoes. So, this form stays intact in the question. And, you'd never say size 7 of shoes. So, you would be unlikely to use the form:

What size of shoes do you wear?

It just sounds awkward (at least to a speaker of AmE).

As, FumbleFingers rightly states in comments above: much of this is related to your "inner grammarian".