Geometric scaling is well confirmed for transverse-momentum distributions observed in proton-proton collisions at Large Hadron Collider (LHC) energies. We introduced multiplicity dependence on a saturation momentum of the geometrical scaling, assuming the scaling holds for semi-inclusive distributions as well as for inclusive distributions. The saturation momentum is usually given by Bjorken's $x$ variable, but redefinition of the scaling variable can make the saturation momentum a function of collision energy $W$. We treat the energy as a free parameter (denoted $W^{*}$ to distinguish it from $W$) and associate the energy-dependent saturation momentum $Q_{\mathrm{sat}}\left(W^{*}\right)$ with particle number density. By using $Q_{\mathrm{sat}}\left(W^{*}\right)$ for a scaling variable $\tau$, we show semi-inclusive distributions can be geometrically scaled, i.e., all semi-inclusive spectra observed at $W=0.90$, 2.76, and 7.00 TeV overlap one universal function. The particle density dependencies of mean transverse momentum $\langle p_T\rangle$ for LHC energies scales in terms of $Q_{\mathrm{sat}}\left(W^{*}\right)$. Furthermore, our model explains a scaling property of event-by-event $p_T$ fluctuation measure $\sqrt{C_m}/\langle p_T\rangle$ at LHC energies for $pp$ collisions, where $C_m$ is a two-particle transverse-momentum correlator. Our analysis of the $p_T$ fluctuation makes possible to evaluate a nonperturbative coefficient of the gluon correlation function.