From the EEOC:
[E]mployers may take steps to determine if employees entering the workplace have COVID-19 because an individual with the virus will pose a direct threat to the health of others. Therefore, an employer may choose to administer COVID-19 testing to employees before they enter the workplace to determine if they have the virus.
Just because employers can test doesn't mean they should test. Why shouldn't an employer test even though the law allows for it?
1/ Because these tests only measure a small slice of time, and therefore are not a reliable indicator of whether an employee is bringing COVID-19 into the workplace. Unless you are going to test every employee every day (see no. 2 below for why this isn't feasible or practical), a negative test only establishes that an employee does not actively have the COVID-19 virus within their system at the time the test was administered.
2/ Because tests aren't necessarily reliable. A team from John Hopkins Medicine has concluded that COVID-19 tests have a false negative rate of at least 20% if used too early after infection. If one in five employees who tests negative for COVID-19 is carrying the virus, why test at all?
3/ Because tests are expensive and not available in a large enough quantity. Testing every employee every day will cost an employer a small fortune, while at the same time unnecessarily using testing resources that could be put to better use (i.e., for those who actually have COVID-19 symptoms). It's for this same reason (among others) that I believe restarting professional sports right now is crazy. Those players are being tested every single day, using up valuable testing resources that are in too short supply as is.
What should an employer do instead of testing? All employers should require employees to self-monitor for COVID-19 symptoms, and upon the presentation of any such symptoms, report to the employer and isolate per CDC guidelines. Thereafter, the employer should contact its local department of health, contact trace, and quarantine those who were in close contact as needed. That's about the best you can hope to do to help stop this virus from spreading in your workplace. The law allows you to require testing if you want, but why bother?