Can I power ICs directly from a wall adapter?
The main distinguishing factor of audio circuits is the low noise level that must be maintained.  While powering a audio circuit from some wall wart will work, it may, and quite possibly will, result in unacceptably high noise.

Most wall warts today are switchers.  The switching frequency is usually well above the audible range.  However, depending on the control algorithm, there may be sub-harmonics in the audio range.

With a sufficiently strong and low impedance input signal, a little extra noise from the power supply might not matter.

You haven't said anything about the circuit you are powering.  The circuit may have some power supply rejection.  If the circuit is opamp-based, then this is probably true.  However, active power supply rejection only works up to some frequency.  Above that, the active circuit can't keep up with, and therefore compensate for, the power supply noise.  The switching frequency of a common wall wart may well be above that range.  You can't hear the 1 MHz or whatever switching frequency, but such high frequency noise inside a chip can mess up its normal operation on much lower frequencies.  The noise could get AM-demodulated and added to your signal, for example.

All in all, I think it's a bad idea for "audio" without knowing something more specific about the input signal, output signal, and the circuit.

If your circuit contains a power stage, then the worst power supply noise could be from the changing current demand of the power stage.  If this is not dealt with properly, then your circuit could oscillate.

It would be prudent to filter the supply to any sensitive front end circuit.  You can use the raw unfiltered supply for a power stage.  Note that sensitive front end circuitry generally requires little power current, so a few ferrite chip inductors followed by capacitors to ground won't get in the way.

Medical grade or not has no bearing on audio use.  Medical grade means extra low leakage between power input and output, and possibly extra high isolation voltage.  It has nothing to do with noise on the output.