This isn't so much of a problem as it appears on the surface. In point of fact, this may actually result in economic growth.
The first thing that's important to note is that the number of days lost to illness by employers goes down to zero. That means that most business are already more effective because you don't have to worry about sick leave for anything. Even accidents and murders are more or less extinct, so training new staff doesn't carry the risk of loss to injury or death. Loss to competitors is still an issue, but overall your training dollars are worth more productivity than they are in the past because the risk of losing the skills developed actually goes down.
Sure, it can be argued that some employers are going to need less staff because they no longer have to over-hire to account for sick leave among their number, but the number of working days available to the employers has just gone up. This means that despite any short term adjustments that some companies will doubtless engage in, the economy is likely to benefit from a more productive workforce in the medium to long term.
As for the healthcare industry; sure there are going to be losses here. Nurses in particular are going to be hit hard as their entire career is built around the idea of caring for the sick. That said, their skills are still quite formidable and their concern for others make them ideally suited in some other industries, like occupational therapy and Workplace Health and Safety assessments.
Doctors may be better off; some will retrain around psychiatry and psychology (mental health could be a little worse off in this environment due to the strain of not being able to take sick days to get some rest away from the stress of work) and medical receptionists and administrators just take their skills to other industries. The rest of the GPs and the like still have analytical and diagnostic capabilities so they will find themselves able to retrain into other STEM fields which (while inconvenient) isn't fatal to aspirations of a rewarding career.
Hospitals can be re-purposed as buildings, and some measure of aged care will still be required as well so at least some of the beds will need to be kept. The OP hasn't discussed the risks associated with childbirth, but there will still be a need for midwives (even if not obstetricians) to assist with this in an ideal environment. So, maternity wards may not shut down either.
The medical research world (and for that matter the pharmaceutical industry) still have roles to play insofar as ongoing medical research would now focus on longevity rather than organic maintenance. Pharmaceutical companies are already in the chemical industry in most cases (Ever noticed that the company Bayer makes both aspirin and Baygon, an insect surface spray?) so while they may wind back certain production activities, they just focus more on others. More research on fertilizers to feed the growing population for instance.
Even companies that manufacture diagnostic imaging equipment just use their expertise to solve different problems.
In the short term, yes there will need to be an adjustment. Some people will struggle as this change represents a massive redistribution of skills and priorities for society as well as business. But in the medium to long term, the skills that are freed up by the lack of medical demand are still useful skills that could be applied to accelerating the resolution of many different intractable problems that mankind still hasn't solved yet.
That, and the increased average productivity of workers in general, mean a healthier economy over time.
It should be pointed out that this very problem was postulated as a potential outcome of the introduction of computers into the business environment. To be sure, computers DID make some jobs redundant and continue to do so. When was the last time you saw a position vacant for a filing clerk at a registry? That said, in terms of numbers, even more jobs have been created by computers than have been lost and this continues to be the trend. What computers have done is automated the lower skilled jobs and created many more highly skilled roles that need to be filled.
The removal of medicine would result in a similar adjustment but with a singular exception; everyone with a job disrupted by this change is already highly skilled and can enter the workforce almost immediately (or with some additional light training) in another field. In that one respect, this would cause less of a disruption to the modern economy as the introduction of computers has.