The mass of the electron is roughly $10^{-30}$ kg. The mass of the proton iis roughly $10^{-27}$ kg. The distance between them is roughly $10^{-10}$ m. Inserting these into the formula for gravitational binding energy:

$$U=\frac{Gm_pm_e}{r}=\frac{10^{-11}\times 10^{-27}\times 10^{-30}}{10^{-10}}=10^{-58}\textrm{ J}=10^{-40}\textrm{ eV}$$

Our most sensitive detectors (ones looking for axions in the microwave regimes) can detect energies of around $10^{-3}$ eV.

The mass of the electron is roughly $10^{-30}$ kg. The mass of the proton iis roughly $10^{-27}$ kg. The distance between them is roughly $10^{-10}$ m. Inserting these into the formula for gravitational binding energy:

$$U=\frac{Gm_pm_e}{r}=\frac{10^{-11}\times 10^{-27}\times 10^{-30}}{10^{-10}}=10^{-58}\textrm{ J}=10^{-40}\textrm{ eV}$$

The most sensitive energy measurement we have ever made (i.e. the Harvard Tower version of the Pound-Rebka experiment) measured an energy difference of $10^{-11}$ eV.