Nik's answer seems perfectly suited to what the OP was looking for, but there are indeed other generalizations of (complete) regularity to C*-algebras, as shown in Rosický's "[Multiplicative lattices and C*-algebras][1]" Cahiers Top. Geom. Diff. Cat. tome 30 no 2 (1989), 95-110.  To paraphrase part of that paper, note first that regularity can be stated in terms of open and closed sets, without reference to points.  Specifically, a topological space $X$ is regular iff, for every open $O\subseteq X$,
$$O=\bigcup\{N:N\text{ is open and }\overline{N}\subseteq O\}.$$
So a natural question to ask is whether, for any C*-algebra $A$ and open projection $q\in A^{**}$,
$$q=\bigvee\{p\in A^{**}:p\text{ is an open projection and }\overline{p}\leq q\}?$$
The answer is yes.  To see this, take any $a\in A^1_+$ with $a\leq q$ and note that $\overline{a_{(\epsilon,1]}}\leq a_{[\epsilon,1]}\leq a_{(0,1]}\leq q$, for any $\epsilon>0$, where $a_S$ denotes the spectral projection of $a$ in $A^{**}$ corresponding to $S\subseteq\mathbb{R}$.  The result now follows because $q$ being open means
$$q=\bigvee\{a_{(0,1]}:a\in A^1_+\text{ and }a\leq q\}.\\$$

As $\overline{N\cup M}=\overline{N}\cup\overline{M}$, for open subsets $N$ and $M$, above we actually have a directed union.  On the other hand, we can have $\overline{p\vee r}\neq\overline{p}\vee\overline{r}$ for open projections $p$ and $r$, as in Example II.6 in Akemann's paper referred to by Nik, which also shows that the supremum above is not always directed.  However, we might ask:

> Can the open projections above be replaced by a directed subset?

In general I do not know.  If $A$ is separable then the answer is yes, as then every open $p$ will be of the form $a_{(0,1]}$, for some $a\in A^1_+$, and $a_{(\epsilon,1]}$ is directed, for $\epsilon>0$.  More generally, the answer is yes if $A$ has an "almost idempotent" approximate unit $(a_\lambda)\subseteq A^1_+$, i.e. such that $a_\lambda a_\gamma=a_\lambda$ whenever $\lambda<\gamma$.  Wondering if, in fact, all C*-algebras have such an approximate unit led me to post [this question][2].

Regardless, in general we can still get a weaker iterated form of directedness.  Specifically, note that $(a+b)_{(0,1]}=a_{(0,1]}\vee b_{(0,1]}$, for all $a,b\in A^1_+$, so every open projection $r$ is a directed supremum of open projections $q$, which are all themselves directed supremums of open projections $p$ with $\overline{p}\leq q$.

Also note the relation $\overline{N}\subseteq O$ can be expressed purely in terms of open sets as
$$\text{there exists open $M$ disjoint from $N$ such that }X=M\cup O.$$
Wondering if, for open projections $p,q\in A^{**}$, $\overline{p}\leq q$ can also be expressed similarly in terms of open projections just led me to post [this question][3].  In fact, it is really relation 2. of that question, used with iterated directness as above, that Rosický considers as the appropriate notion of regularity for C*-algebras in the paper mentioned above.

For _complete_ regularity one replaces $\overline{N}\subseteq O$ with the statement that there exist open $(O_r)$ for $r\in[0,1]$ (or $[0,1]\cap\mathbb{Q}$, as in Rosický's paper) such that $N\subseteq O_0$, $O_1\subseteq O$ and $\overline{O}_s\subseteq O_t$ whenever $s<t$.  Again, the analogous statement for complete regularity holds for open projections in C*-algebras although again it is not clear if the supremum can always be replaced with a directed supremum.


  [1]: http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1989__30_2/CTGDC_1989__30_2_95_0/CTGDC_1989__30_2_95_0.pdf
  [2]: https://mathoverflow.net/questions/221721/almost-idempotent-approximate-units-in-c-algebras
  [3]: https://mathoverflow.net/questions/221671/closed-containment-of-open-projections-in-c-algebras