Center The Text!

vim, 43 36 35 bytes

VGrx:sor
G:let &tw=col("$")
uu:%ce

Too good not to post. Note the trailing newline; it is significant.

Thanks to @Marth for saving a character!

vim-friendly format:

VGrx:sor<cr>G:let &tw=col("$")<cr>uu:%ce<cr>

Explanation:

VGrx                   replace every character with an "x"
:sor<cr>               sort (since all chars are now same, sorts by line length)
G                      go to the very last line
:let &tw=col("$")<cr>  set &tw to column number of last char on this line
                         "let &tw" is equivalent to "set tw"
                         tw is short for textwidth, used in :center
uu                     undo the sort, and the replacing-with-x too
:%ce<cr>               center over entire file (%), using textwidth set earlier

Mathematica, 96 bytes

StringRiffle[#~StringPadLeft~Floor[Max@(l=StringLength)@a/2+l@#/2]&/@(a=#~StringSplit~"
"),"
"]&

Don't ask me how it worked, I just fiddled with it until it produced the correct output.

Pyth, 19 17 bytes

2 bytes thanks to Jakube

V.ztr+1.[l.T.zNd6

Demonstration

I think this is the first time the center-pad function of .[ has been useful. The length of the longest line is found using non-truncating transpose (.T).

Trailing spaces are removed by adding a non-space character to the front, stripping spaces, then removing the added character.