The answer is yes also for $L_p$, but I don't know a good book reference.  For $2<p<\infty$, this is contained the paper of Kadec and Pelczynski--it is their second dichotomy theorem.  Actually, they get that a normalized weakly null sequence has a subsequence that is either equivalent to an orthonormal sequence (in which case its closed span is automatically complemented) or has, for every $\epsilon > 0$, a subsequence that is $1+\epsilon$-equivalent to the unit vector basis for $\ell_p$ and spans a subspace that is $1+\epsilon$-complemented.

For $1\le p <2$, I think the result was pointed out by Pelczynski but I don't know a reference.  It follows from arguments like those in Wojtaszczyk's book characterizing weak compactness in $L_1$.  You can find an outline of the argument in a paper I wrote with G. Schechtman:

Multiplication operators on L(Lp) and lp-strictly singular operators, J. European Math. Society 10 1105-1119 (2008), which you can download from my home page.

EDIT July 7, 2012: The result above that I attributed to Pelczynski is actually due to Enflo and Rosenthal:

Enflo, Per; Rosenthal, Haskell P.
Some results concerning Lp(μ)-spaces. 
J. Functional Analysis 14 (1973), 325–348.