Blackadar's [K-Theory for operator algebras][1] 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 they show similarity by showing the stronger property of homotopy equivalence.  It is [Proposition 4.6.2][2] on page 23 of the 2nd edition (1998).  ([Proposition 4.3.3][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][4], with a very similar proof.


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Added after the first two comments:

Kaplansky's *[Rings of operators][5]* 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.)  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.


  [1]: http://books.google.com/books?id=214a1Wri63QC&dq=idempotent+projection+similar&client=firefox-a&source=gbs_navlinks_s
  [2]: http://books.google.com/books?id=214a1Wri63QC&lpg=PA23&dq=idempotent%2520projection%2520similar&client=firefox-a&pg=PA23#v=onepage&q=&f=false
  [3]: http://books.google.com/books?id=214a1Wri63QC&lpg=PA23&dq=idempotent%2520projection%2520similar&client=firefox-a&pg=PA21#v=onepage&q=idempotent%2520projection%2520similar&f=false
  [4]: http://books.google.com/books?id=SMiB8VIB5UIC&lpg=PA192&dq=idempotent%2520projection%2520k-theory&client=firefox-a&pg=PA192#v=onepage&q=&f=false
  [5]: http://books.google.com/books?id=hRaoAAAAIAAJ&client=firefox-a&source=gbs_navlinks_s