Rather than reading the wall of text above, I am basing my reply off of [the answer already provided][1]:

The Arnold proof is well known to be erroneous, but a correct (as far as I know) version is cited in an earlier MO post [here][2]. In particular, it is a proof of the FTA via the Brouwer Fixed Point Theorem.

The latter source is: 

Some Properties of Continuous Functions. M. K. Fort, Jr. The American Mathematical Monthly, Vol. 59, No. 6 (Jun. - Jul., 1952), pp. 372-375. http://www.jstor.org/stable/2306806.

[**Edit:** Todd Trimble has kindly provided a link to the Fort paper that does not require jstor access.]

Separately, I see the following quotation:

"Recently, there have been very interesting proofs of the Brouwer theorem. Kulpa deduced a generalization of the Brouwer theorem from the Fubini theorem and the Weierstrass approximation theorem, and applied it to give a simple proof of the fundamental theorem of algebra."

The source of this excerpt is: 

Park, S. (1999). Ninety years of the Brouwer fixed point theorem. Vietnam Journal of Mathematics, 27(3), 187-222. http://www.math.ac.vn/publications/vjm/vjm_27/No.3/187-222_Park.PDF

And the reference under discussion is:

W. Kulpa, An integral criterion for coincidence property, Radovi Mat.6 (1990) 313-321.

I gathered this information at the request of D. Goroff some time ago, at which point my search for the Kulpa paper was, unfortunately, fruitless. **If anyone can find an accessible copy of this paper, I would be most interested in it (and I know he would be as well).**


  [1]: http://mathoverflow.net/questions/132036/can-you-prove-the-fundamental-theorm-of-algebra-just-using-fixed-point-theory/132038#132038
  [2]: http://mathoverflow.net/questions/19272/applications-of-brouwers-fixed-point-theorem/112779#112779