Yes. You can see this directly with the rank-nullity theorem: if $P$ has rank $m$, then because $P^2=P$ you get that every vector in the image is an eigenvector with eigenvalue $1$. Here's the proof: if $b \in C(P)$, then $b=Px$, now $Pb=P^2x=Px=b$, so $b$ is an eigenvector with eigenvalue $1$.
So the eigenvalue $1$ has geometric multiplicity $m$. Then the rank-nullity theorem tells you the nullity is $n-m$, i.e. the geometric multiplicity of the eigenvalue $0$ is $n-m$.
Note that this actually does not depend on $P$ being an orthogonal projector, it only requires $P^2=P$.