Not a definitive answer, but some pointers which are too long for a comment.

Let me just check I'm correctly interpreting the question. You take a uniformly random $r$-regular graph on $2n$ vertices. Then (independently of the edges) you uniformly colour $n$ vertices white, and consider the subgraph spanned by those $n$ vertices. Now you want to look at the largest component of that graph on $n$ vertices.

In that white graph, consider a typical vertex, and consider its neighbours, the neighbours of those neighbours, etc. As $n\to\infty$, what you see locally converges to a branching process with offspring distribution Binomial($r-1, 1/2$) (except for the first generation which is Binomial($r,1/2$)).

For $r\geq 4$ this branching process is supercritical -- the mean family size is $(r-1)/2>1$. In this case one would expect a "giant" component in the white subgraph, of size $\sim \theta_r n$ for some constant $\theta_r$. In fact $\theta_r$ should be the survival probability for a branching process of the type described in the previous paragraph. The second-largest component should be of order $\log n$.

For $r=3$ the process is critical. By analogy with other familiar classes of critical random graphs, one would expect the largest component to be on the order of $n^{2/3}$. (Also the $k$th largest component will be on the order of $n^{2/3}$ as $n\to\infty$ for any fixed $k$.)

I think this should all be doable within the context of known results for "configuration models". The empirical degree distribution of the white subgraph converges to Binomial($r$, $1/2$), and given the degree sequence, the graph is uniform among all possibilities -- i.e. it's a configuration model conditioned to be simple. You can find copious information and references about the structure of configuration models (including questions about large components) in volume 2 of Remco van der Hofstad's book Random Graphs and Complex Networks. An example of a reference to cover the critical ($r=3$) case might be Critical window for the configuration model: finite third moment degrees by Dhara, van der Hofstad, van Leeuwaarden and Sen (but that might also be overkill -- perhaps there are earlier and simpler results which would already do the job).