Although it isn’t a physiological process internal to the human (or animal body), one of the very significant things gravity does for us is ensures that the physical vectors of disease (referred to in hygiene literature as “fomites” or “fomes”) are limited in their travel and persistence in the environment. In a terrestrial environment with normal gravity, carriers like mucus, pus, blood, excrement, and other expectorants and bodily fluids which provide protection, nutrition, and transport for viral and bacterial vectors will settle on horizontal surfaces which can be readily sanitized. This substantially limits the spread of such diseases to a manageable level. In freefall, however, fomites will remain suspended in the air or will stick to surfaces in every direction, increasing their distribution and persistence. And even debris that does not carry disease vectors is still problematic, as John Young demonstrated with his “contraband” corned beef sandwich on Gemini III with concerns about the crumbs from the bread causing electrical shorts or blocking the HVAC systems. In current habitat systems, this is dealt with by having constant air flow and filtration to remove contaminants, but it doesn’t prevent vectors from attaching to surfaces, especially where air flow tends to stagnate. The infamous black mold on the Soviet/Russian Mir station, which threatened to make the station uninhabitable prior to its destruction is the most notable example, but the general problem will remain an issue in any freefall habitat.
njtt notes the long term physiological problems with living in freefall conditions. Our longest continuous experience with people in freefall conditions is about 14 months (Valeri Polyakov) so we don’t have a comprehensive model of human tolerance and variability to long term low or zero effective gravity conditions. (Yes, I am aware that freefall in orbit is not technically “zero gravity” but as far as the subject is concerned there is no acceleration field in their orbiting frame of reference.) However, the more we learn about the response of human physiology to freefall conditions and the radiation environment in unprotected space, the more apparent it is that in order to support long term human habitation we will have to simulate terrestrial conditions to a substantial degree of fidelity, e.g. radiation protection, Earth-like atmosphere and conditions, a significant simulated gravity field. So while there are no essential physiological functions that are immediately dependent up on gravity, in the long term we are not “built” for space travel any more than we are adapted to swim in the ocean continuously.