This has been asked on quora, here are the different aspects mentioned:
The simple answer is E, as the smallest, lightest string, is more susceptible to being pulled back out of tune by the tuning of the other three heavier strings changing the tension on the neck of the violin.
By tuning the others first, then tuning E, you avoid needing to retune E.
Basically, the tuning of the lower strings could put the E out of tune, so we end with it.
Graham C Lindsay:
Its to do with comparisons and your ears. The lower strings have lower harmonics and if you start from the bottom to the top with the other strings tuned when you get to the e string it will be easier to tell if the string is in tune.
You could try it the other way but often you will find that it takes longer to tune the instrument as you may have to keep adjusting the other strings.
This is a much better explanation than what I was trying to say. Having lower strings resonate along with the E indicates where really we have to set it.
We pretend that our tuning system works, but it only almost works. An octave is based on a 2:1 Hz ratio. If open A is 440 cycles per second, then A on the G is 220. A perfect 5th is 3:2, so E on the D is 330. If you start with the lowest note on the piano and double the frequency until you get to the top A on the piano, you will get a certain number. Start with the same low note, and do 3:2 5ths until you get to the top of the piano, and the number will be about a half step sharp. (A half step is 15:16). When tuning in 5ths, to make your music match the rest of the orchestra and/or piano, you will have to “squeeze” the fifths to make things work. Tune to a perfect plain-sounding 5th and then make the fifth a little smaller. This can be more of an issue on the viola and cello. When tuning my viola, I like to tune the A and the C against the piano, and then tune the D and G so that each 5th sounds the same amount off. In a quartet, I check my C against the cello C to make sure we both squeezed the same amount.
This is a bit orthogonal to the question, but it really is important to mention whenever talking about violin tuning.