I'm not opposed to that. Would need to have a limited number of unverified sick days as Americans would abuse it, at least for a time. If you go to a doctor and they say it is necessary for you to be off work for 3 months, then an employer would have to respect that. There would also need to be punishment for retaliating against employees that have long sick periods.
One compromise is businesses could be compensated by the government for sick time employees use beyond what is given by company policy. If someone has to take two weeks and they were only slotted one, the government will give a fixed amount per day that does not exceed that employee's daily wages.