What is the minimum amount of time that a salaried exempt employee must work to be paid for the entire day?
If an exempt employee does any work, they must be paid for the full day - there is no minimum. For example, if an employee worked for the first 20 minutes of their usual 8-hour day, then had to go home to deal with a flooding basement, they would be entitled to their full pay for that day.
You may, however, have the option of using an employee's paid time off to fill in the gap. If the employee has hours available in their PTO or vacation bank, you could use time from that bank to cover their absence (Assuming this is your standard policy and practice). If, however, the employee was out of paid time off - or was never offered any- you would still have to pay them for the full day. You should also be careful about any state-mandated sick leave to fill in the gap, even if the employee is sick; employees often get to choose when to use this leave.
If you would prefer to pay employees only for the hours they work, you could reclassify your salaried exempt employees as hourly non-exempt. Of course, non-exempt employees are entitled to overtime.