No. Akemann has constructed a C-algebra that does not contain an approximate unit of commuting elements. In particular, this CC*-algebra that does not contain an almost idempotent approximate unit of commuting elements. See Example 2.1 in
Akemann. Approximate units and maximal abelian C*-subalgebras. Pacific J. Math. 33 (1970)
As pointed out by Tristan, a C*-algebra might have no approximate unit of commuting elements but nevertheless have an almost idempotent approximate unit. (Thanks for pointing out the mistake in my original answer.) So the original question remains open.