How can US faculty verify medical absences?

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.

I don't know about the US, but in the UK and Australia it is standard practice to request a doctor's certificate (same as for sick leave from employment, e.g., https://www.fairwork.gov.au/leave/sick-and-carers-leave/paid-sick-and-carers-leave/notice-and-medical-certificates). This is not the same as obtaining medical records. Typically these are checked by administrative staff to verify the doctor actually administered it, since students have been known to forge them.

As stated in multiple comments, ask the Division of Student Affairs, the Dean of Students, or similar organization, to verify the student absence for you.

Having this intermediate will save you from violating your student's privacy, and also relieve you from the burden of assessing whenever a medical certificate is genuine or not.