Reading this question What happens to an electron in a molecule once it has absorbed a photon and transitioned? it occurs the question to me is the ground state say of a hydrogen electron the only one? Why I ask? Because under normal conditions by 24°C the electron is exposed by thermal radiation, means it is influenced by EM radiation all the time. So does the ground state depends from the surrounding temperature and to be exact THE ground state has to be mentioned always with the temperature for which is it meant? The same holds for the gravitational potential?
When we talk about the ground state of hydrogen we generally mean the lowest energy eigenfunction of the time independent Schrodinger equation. Strictly speaking no hydrogen atom is ever in that state because time independence means it would have had to be in that state for an infinite time and continue in that state for an infinite time into the future. However under most circumstances this is an unnecessarily pedantic viewpoint.
A real hydrogen atom is bathed in a sea of EM radiation - even if floating in space it interacts with the cosmic microwave background. This will indeed perturb the ground state and we can calculate the effects using perturbation theory. However this doesn't have any significant effect unless the energy of the radiation is large enough to stimulate a transition. On average the ground state remains so close to the theoretical ground state as to be indistinguishable.
One example of where environmental effects are important is in very strong magnetic fields where the ground state can be significantly altered. We can shift the energies of the eigenfunctions in the lab by applying magnetic fields, and we expect that there are natural examples like neutron stars where the magnetic filds are large enough to have a big effect on the atomic states.
It isn't obvious what you mean by gravitational interactions. A hydrogen atom would only be affected by tidal forces, and on the atomic scale these are negligably small unless the poor atom happens to be right next to a black hole singularity.