Somewhat, it depends on your location.
In the US, OSHA sets out rules and regulations for safety and workplace health and I believe they do a fair amount of oversight to enforce certain health standards. Companies in the US have to abide by this. I think in most companies, the experts at meeting these standards would probably be the Facilities group/division/organization that is in charge of maintaining the physical work environment, as most of these regulations have to do with having a safe and healthy physical space. There are other best practices and evolving science around how to make office environments better for humans - and most of the time what I see is that this is something that the physical facilities groups work on.
However, there's some level of crossover into Human Resources if there's a time when health regulations/best practices cross over into how the company treats its employees. For example, I could see HR getting involved if the company invented new policies around hand-washing that involved a change in how warnings and rule enforcement is conducted - then HR and even legal may be involved in the change. Similarly, since HR typically runs employee benefits including medical care coverage (unless in your country, it's covered by the government) - then the balance between employee health needs and medical coverage is managed by HR. For example - at one point I escalated through HR when I was simultaneously getting company guidance to make sure that everyone was vaccinated and/or immune to chicken pox, and yet my medical insurance provider had refused to pay for vaccination. They actually interceded and made sure it was covered. Certainly that fits as both a personal AND public health concern!
Next - some snazzy tech companies (this really is a perk you only see in bigger/richer companies) - have an Ergonomics group. Usually they tie into facilities as a lot of what they do involves alterations to the physical environment to make it better for the workers. This the technology/physiology of making sure that activities that are done by employees are done in the least harmful way possible. At least in my company it is stuff like correct desk/chair alignment, keyboard/keyboard trays, and adjusting for glare, etc.
There's yet more regulation and probably other parts of a company that come into play if the company's outputs in some way affect public health. For example, restaurants have to be licensed and inspected to make sure they are handling their food in a safe way in terms of food-born pathogens (not so much for how healthy the food may be in terms of calories or saturated fats & sugar!).
And I think that there are also environmental laws and regulations and oversight there although there seem to be plenty of terrible outcomes that suggest the world could do a lot better here.
Lastly, there is usually some guidance given to management on how to help manage work/life issues with employees, how to work with cases of disability and accommodating reasonable requests for support, how just generally be an ethical and humane manager. This all hits both health concerns and legal concerns.
Those things all get more clear the bigger your company - my experience is with 8,000 person and larger companies. When you have something like a 200 person company, you may not see any of this, as you just don't have the structure or the need for it. They also probably don't have the slack and/or economy of scale to do much in terms of larger public health concerns. And even in a big company - different companies prioritize different things - some try to be very green, and there are all sorts of green/sustainable building initiatives that are sometimes optional, others try to have a great record in terms of employee health, others try to have a better impact on the whole community around them - in some ways, it fits into how the company intends to brand itself.