Ground-state pair potentials between protons immersed in jellium (interacting electrons plus a compensating background) and at densities corresponding to the range $1<{r}_{s}<4$ are calculated using first-principles density functional theory methods. While the results obtained for immersing a single H atom in an electron gas agree with previous calculations, it is discovered that molecular-type short-range binding becomes unstable at surprisingly low density (at about ${r}_{s}=3.2),$ and that at even lower density there is a bistability between a weakly bound molecule and an unpaired state. This behavior is a result of the possible pairing of hydrogen and itinerant jellium electrons with little electrostatic penalty, a consequence of the positive background filling the entire volume V. Another important finding is a density range $(1.5<{r}_{s}<2)$ where the pair potentials are insufficiently strong to bind the protons, suggesting the possibility of inducing hydrogen into a state of low-temperature quantum fluid at a critical density.