Blackadar's K-Theory for operator algebras has it, although the way it is done there is perhaps overkill if this is all you need. The result is generalized to local $C^*$-algebras, and he shows similarity by showing the stronger property of homotopy equivalence. It is Proposition 4.6.2 on page 23 of the 2nd edition (1998). (Proposition 4.3.3 shows that homotopy equivalence is stronger.)
The stronger equivalence (but just for $C^*$-algebras) is also shown in the K-theory book by Rørdam et al., Lemma 11.2.7, with a very similar proof.
Added after the first two comments:
Kaplansky's Rings of operators has another approach. Since there is no Google preview, I'll outline what is done. Theorem 26 shows that if $A$ is a unital ring with involution $*$ such that $1+x^*x$ is invertible for all $x\in A$, then for each idempotent $f\in A$ there is a projection $e\in A$ such that $fA=eA$. (The projection is obtained just as in the previous two sources, $e=ff^*(1+(f-f^*)(f^*-f))^{-1}$.) A previous exercise (4 on page 24) shows that if $f$ and $e$ are idempotents in a unital ring $A$ and $fA=eA$, then $f$ and $e$ are similar.