Automatically stretch table to evenly fill horizontal space?

Other answers have shown how to use tabularx however your description was mistaken, tabularx never changes \tabcolsep. However if your sample data is typical I think you do want the inter-column space to stretch and allow the column widths to be based on the natural column widths. For this you want the standard LaTeX tabular* not tabularx.

\documentclass{article}
\usepackage{showframe}% http://ctan.org/pkg/showframe
\usepackage{booktabs}% http://ctan.org/pkg/booktabs
% tabularx already includes the array package
%\usepackage{array}% http://ctan.org/pkg/array

\begin{document}
\noindent\begin{tabular*}{\columnwidth}{@{\extracolsep{\stretch{1}}}*{7}{r}@{}}
  \toprule
  & $z_{6}$ & $z_{8}$ & $z_{9}$ & $z_{11}$ & $z_{13}$ & $z_{14}$ \\
  \midrule
  fileA & 0.00 & 0.00 & 0.00 & 0.08 & 0.79 & 0.08  \\
  fileB & 0.01 & 0.00 & 0.13 & 0.00 & 0.84 & 0.00  \\
  fileC & 0.00 & 0.39 & 0.02 & 0.49 & 0.00 & 0.00  \\
  fileD & 0.75 & 0.08 & 0.00 & 0.00 & 0.00 & 0.00  \\
  \bottomrule                             
\end{tabular*}
\end{document}​

You're not using the tabularx environment "properly". tabularx provides an X column type that stretches as necessary. In order to obtain a right-aligned X column, you can use

\newcolumntype{R}{>{\raggedleft\arraybackslash}X}

Here is a minimal example showing your full-width table:

\documentclass{article}
\usepackage{showframe}% http://ctan.org/pkg/showframe
\usepackage{tabularx}% http://ctan.org/pkg/tabularx
\usepackage{booktabs}% http://ctan.org/pkg/booktabs
% tabularx already includes the array package
%\usepackage{array}% http://ctan.org/pkg/array
\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
\begin{document}
\noindent\begin{tabularx}{\columnwidth}{ *{7}{R} }
  \toprule
  & $z_{6}$ & $z_{8}$ & $z_{9}$ & $z_{11}$ & $z_{13}$ & $z_{14}$ \\
  \midrule
  fileA & 0.00 & 0.00 & 0.00 & 0.08 & 0.79 & 0.08  \\
  fileB & 0.01 & 0.00 & 0.13 & 0.00 & 0.84 & 0.00  \\
  fileC & 0.00 & 0.39 & 0.02 & 0.49 & 0.00 & 0.00  \\
  fileD & 0.75 & 0.08 & 0.00 & 0.00 & 0.00 & 0.00  \\
  \bottomrule                             
\end{tabularx}
\end{document}​

The inclusion of showframe is merely to show that the table spans the entire column width.

Also note the reduced column specification when you're using similarly-specified columns: *{<num>}{<col spec>} duplicates <col spec> a total of <num> times. The newly-defined column type R inserts \raggedleft before each cell entry, pushing the contents flush right.