How can US faculty verify medical absences? I recently received a memo stating that I am not allowed to ask for doctor's notes when students are absent from class, as this is allegedly a violation of privacy rights.

If this is the case, then how can a faculty member legitimately verify that a student who claims to have missed an exam or other activity as a result of illness was, in fact, sick as claimed? Couldn't students abuse such a policy to postpone an assessment for which they feel unprepared?
Couldn't students abuse such a policy to postpone an assessment for
  which they feel unprepared?


Yes. My experience teaching the first semester at a community college is that granted an open-door absence policy, the majority of my students were skipping all the tests, circulating the real test among themselves, and then taking a makeup together on a later date. (Assuming I could get them all in one place.) This was more than double my scheduling/test-making/test-grading labor, and also delaying the assessment/feedback cycle by a week or more for each test. 

In my second semester I ended that policy. The policy since has been: Nothing is excused for any reason. No excuses or notes need be given to me. One low test score is dropped (and likewise for any other assignments). This is an enormous time-saver (both in test-giving and excuse-verification), and allows me to immediately hand back corrections and feedback in the next class meeting.