Empirical Relation of the Gravitational Constant from $c G \hbar H_0$ Physics

We propose a novel empirical relation for the gravitational constant $G$ that incorporates the speed of light $c$, reduced Planck constant $\hbar$, the Hubble parameter $H_0$, the proton mass $m_p$, the elementary charge $e$, and the vacuum permittivity $\varepsilon_0$. The formula $G = \frac{c^2 H_0}{\sqrt{2} \hbar} \frac{4}{3} \pi \left( \frac{\hbar^2\varepsilon_0 }{m_p e^2} \right)^3$ yields a value within 0.037\% of the CODATA 2018 measurement when using the CMB-inferred $H_0$. This relation suggests potential connections between quantum, electromagnetic, relativistic, and cosmological scales, offering insights into unification frameworks amid the Hubble tension.