I believe the answer is yes. There are perhaps several ways to see this, but a decent reference is the book "Affine Flag Manifolds and Principal Bundles" by Schmitt. It contains an article called "Quantization of Hitchin’s Integrable System and the Geometric Langlands Conjecture" by T.L. Gomez. Your answer can be found near the beginning of Section 4.1.
I would just caution that the affine Grassmannian is a projective ind-scheme, while a Whitney stratification is usually done on a manifold or smooth variety. To address this potential issue, I would suggest choosing an embedding $G\hookrightarrow GL_n$ of $G$ as a Zariski-closed subgroup. This allows you to form smooth projective subvarieties $Gr_n$, where $n$ denotes the maximum allowed pole order. These subvarieties are $G(\mathbb{C}[[t]])$-invariant, and hence each is a union of $Gr^{\lambda}$'s. It might be reasonable to first think of the $Gr_n$'s as having a Whitney stratification into $Gr^{\lambda}$'s. Noting that the $Gr_n$'s form a filtration of $Gr$, you can then gain insight into the Whitney stratification of $Gr$.