Use of Concentration for pH
Current definition implies that $\mathrm{pH}$ is a function of relative activity.
Originally, the amount concentration of  $\ce{H+}$ in $\pu{mol L-1}$ was proposed, which is also often used these days as an approximation [1].

$\mathrm{pH}$ was originally defined by Sørensen in 1909 … in terms of the concentration of hydrogen ions (in modern nomenclature) as $\mathrm{pH} = −\lg(c_\ce{H}/c^\circ)$ where $c_\ce{H}$ is the hydrogen ion concentration in $\pu{mol dm–3},$ and $c^\circ = \pu{1 mol dm–3}$ is the standard amount concentration. Subsequently …, it has been accepted that it is more satisfactory to define $\mathrm{pH}$ in terms of the relative activity of hydrogen ions in solution
$$\mathrm{pH} = -\lg a_\ce{H} = -\lg (m_\ce{H}γ_\ce{H}/m^\circ)\tag{1}$$
where $a_\ce{H}$ is the relative (molality basis) activity and $γ_\ce{H}$ is the molal activity coefficient of the hydrogen ion $\ce{H+}$ at the molality $m_\ce{H},$ and $m^\circ$ is the standard molality. The quantity $\mathrm{pH}$ is intended to be a measure of the activity of hydrogen ions in solution.
However, since it is defined in terms of a quantity that cannot be measured by a thermodynamically valid method, eq. 1 can be only a notional definition of $\mathrm{pH}.$

So, if the numerical value for normality coincides with the one for molarity, and one can omit the activity in favor of molarity, i.e. when both equivalence factor $f_\mathrm{eq}$ and activity coefficient $γ$ are approx. 1, then yes, normality can also be used to estimate the $\mathrm{pH}.$
I'm not sure how to interpret &quot;present by itself&quot; part of your question, but if it means there is a single solute, then it seems to be incorrect.
The simplest counter example would be a solution of di-, tri- and polybasic acid/base, say $\ce{H2SO4},$ for which the equivalent concentration is double the value of amount concentration.
References

Buck, R. P.; Rondinini, S.; Covington, A. K.; Baucke, F. G. K.; Brett, C. M. A.; Camoes, M. F.; Milton, M. J. T.; Mussini, T.; Naumann, R.; Pratt, K. W.; et al. Measurement of pH. Definition, Standards, and Procedures (IUPAC Recommendations 2002). Pure and Applied Chemistry 2002, 74 (11), 2169–2200. DOI: 10.1351/pac200274112169. (Free Access)