Sort of. There is no legal requirement for any time off in the US. But PTO is a very common arrangement.
It is what it sounds like: Paid Time Off. Basically you have some hours available your will be paid for but you don't have to work. You earn PTO at a certain rate for every hour you do actual work, so you have to earn it.
You don't have the right to use them whenever you want; that's up to your boss, usually. You usually have to trade any kind of time off for your PTO: vacation, holidays (even mandatory ones), sick days, etc. There is usually a minimum amount of hours you have to take at a time, so you can't always take an hour here and there, but sometimes you can agree to make up time with you boss.
For extended time off, like if you have surgery, you can usually work with your employer to use a reduced rate of PTO to make it last longer, like get 40% pay. When it runs out, you don't get paid. You may still be able to keep your job, within the rules of FMLA (12 weeks), but you have no income. I am not sure what happens to insurance at that point, but I think you're covered as long as you're employed and pay premiums.
Of course this is not legally required and it's rare that it's even offered if you're part time or work for a small company, since those aren't required to offer any benefits.
I don't work in HR, I'm not a lawyer or doctor, my experience is my own, this is not legal advice, your state laws may be different, etc, etc. YMMV.