US Academia and sick days
For full-time faculty, the question doesn't really apply well because faculty aren't paid hourly or daily. In the departments I've been involved with, the policy is that faculty handle their own classes and minor illnesses. That is, if I need to cancel class one day then what that means for the class is up to me. Maybe I have a TA teach it, or I offer a recorded lecture to fill in time, or even just cancel the class and adjust material accordingly.
What it comes down to is whether I can meet the obligations for the class while handling my illness. For major illnesses where that isn't true, the department would schedule someone else to teach and I would look at my short-term disability benefits, which are commonly available in large US universities. The unfortunate exception is an adjunct faculty member, who simply teaches contract-to-contract, and may not even have benefits through the university.
At my US institution,
Hourly employees (administrative assistants, groundskeepers, custodians, etc.) fill out (electronic) time cards showing all of the hours that they've worked and the hours that they've taken off as vacation or sick leave. They earn hours of sick leave and vacation for each pay biweekly period and can only take paid leave if they have enough time "saved up" (accrued) to cover their absence.
Salaried professional staff (typically employees with college degrees such as computer systems administrators, accountants, etc.) and full-time academic administrators (deans and vice presidents and the like) also accrue leave but use a different set of forms to track their leave balances.
Faculty on 9-month contracts do not accrue sick leave or vacation and do not generally have to account specifically for how (or where) they spend their time. However, they are required to teach their classes and be available for students to talk to for at least a few hours a week outside of class time. They can take vacation whenever they want when classes are not in session (summer, Christmas break, and spring break.) If they get sick, they stay home and try to make alternate arrangements for teaching their classes (e.g. a grad student TA or another faculty member might cover their teaching for a few days.)
For serious and long-lasting illnesses that would keep an employee out for many weeks or months, all three groups have disability insurance that pays an employee's salary while they can't work. There's also a separate type of insurance called "workman's compensation" that specifically covers situations in which an employee is injured in the course of their work.
The kind of freedom that faculty have to manage their time is very uncommon in corporate jobs in the US, although self-employed professionals often have much the same freedom.