Question: My question is simple:

Given an algebraic group $G$ acting on a variety $X$ algebraically. If the orbits are of finite number then they form what is called an algebraic stratification of $X$.

Now my question is: **Is this (the stratification by $G$-orbits) also a Whitney Stratification?**

I think showing this by hand is quite painfull since the definition of a Whitney stratification is very unhandy. 

I would appreciate any reference.

Answer: 
So Ulrich and Geordie were right, Tom Braden was the right person to ask and here is what he told me:

The answer is yes, in the case above $X$ is Whitney stratified.

The argument goes roughly as follows. In the paper [K] the following is shown. Let $X$ be an algebraic stratified variety and $ S$ a stratum. The set $Sing(S )$ of points in $S $ which do not fulfil Whitney's b condition has the structure of a (semi-)variety of dimension strictly lower that $\dim S $. 

Now applied to our situation :
$Sing(S)$ is necessarly $G$-invariant, hence empty. This answers my question. 

[K] ``A Geometric Proof of Existence of Whitney Stratifications’’, Moscow Math. Journ., 5 (2005), no.1, 125—133
See:

MR0342523 (49 #7269)
Luna, Domingo
Slices étales. (French) Sur les groupes algébriques, pp. 81–105. Bull. Soc. Math. France, Paris, Memoire 33 Soc. Math. France, Paris, 1973.